Study of the photon structure at LEP

Habilitation Thesis

Mariusz Przybycien

University of Science and TechnologyAl. Mickiewicza 30, Krakow, Poland

Abstract

Two photon physics has become one of the most active fields of research at the recentlyclosed e+e− collider LEP. The measurements of the photon structure functions and of thestructure of interactions of two virtual photons have been performed at LEP using the processe+e− → e+e−γ(⋆)γ(⋆) → e+e− X, where X represents a pair of leptons or a hadronic final state.The results obtained by the LEP experiments in single and double tagged measurements arereviewed and compared with similar results available from other e+e− colliders. In particularthe measurements of the QED and hadronic structure functions of the quasi-real photon aswell as the studies of the possible BFKL effects in the intaractions of two virtual photonsare discussed in detail. The theoretical framework, needed to understand the experimentalresults presented in the paper, is given first.

Contents

1 Introduction 1

2 Kinematics 7

3 Cross sections for the process e+e− → e+e− f f 10

4 Deep inelastic eγ and ee scattering 15

4.1 Photon structure functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 QED structure functions of the photon . . . . . . . . . . . . . . . . . . . . . 19

4.3 Hadronic structure functions of the photon . . . . . . . . . . . . . . . . . . . 21

4.4 Parton distribution functions in the photon . . . . . . . . . . . . . . . . . . . 26

4.5 Hadronic structure of the electron . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Structure and interactions of virtual photons 34

5.1 The cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2 DGLAP and/or BFKL dynamics? . . . . . . . . . . . . . . . . . . . . . . . . 36

6 Monte Carlo generators 40

7 The OPAL detector at LEP 43

8 Studies of the photon and electron structure in DIS 45

8.1 QED structure functions of the photon . . . . . . . . . . . . . . . . . . . . . 45

8.2 Hadronic structure function of the photon . . . . . . . . . . . . . . . . . . . 49

8.2.1 Measurement of F γ2 by the OPAL experiment at low-x . . . . . . . . 49

8.2.2 Measurement of F γ2 by the OPAL experiment at high Q2 . . . . . . . 61

8.2.3 World results on the measurements of F γ2 . . . . . . . . . . . . . . . . 64

i

8.2.4 Charm structure function of the photon . . . . . . . . . . . . . . . . 66

8.3 Hadronic structure function of the electron . . . . . . . . . . . . . . . . . . . 69

9 Interactions of two virtual photons 78

9.1 Leptonic cross section for the scattering of two virtual photons . . . . . . . . 78

9.2 Effective hadronic structure function of virtual photon . . . . . . . . . . . . 78

9.3 Hadronic cross section for the γ⋆γ⋆ scattering process . . . . . . . . . . . . . 80

9.3.1 Event selection and background estimation . . . . . . . . . . . . . . . 81

9.3.2 Comparison with Monte Carlo models . . . . . . . . . . . . . . . . . 85

9.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

10 Summary and outlook 99

A Results on QED structure of the photon 103

B Results on hadronic structure of the photon 107

C Results on hadronic structure of the electron 114

D Results on hadronic structure of the virtual photon 116

References 120

ii

1 Introduction

The idea that energy can be emitted and absorbed only in discrete portions comes fromPlanck and was first presented in the year 1900 in his successful theory describing the en-ergy spectrum of the black body radiation. Only five years later Einstein proposed that lightcan be considered as a flux of particles (light quanta). The notion ‘photon’ was introducedby the American chemist G.N.Lewis in the year 1926.

Over the last century we have observed a huge progress in our understanding of bothmatter and light. The interactions of quarks, leptons and gauge bosons have been success-fully described by the Standard Model – a combination of gauge theories. In this model thephoton plays a role of a gauge boson of quantum electrodynamics (QED) and mediates theelectromagnetic force between charged objects. As the gauge boson of QED, the photon isa massless (m < 2 · 10−16 eV) and chargeless (q < 5 · 10−30e) particle [1] having no internalstructure in the common sense. However, in any quantum field theory, the existence of in-teractions means also that the quanta themself can develop a structure. This follows fromthe Heisenberg uncertainty principle written1 as ∆E∆t > 1. For example, the photon canfluctuate for a short period of time into a charged fermion-antifermion pair, f f, carrying thesame quantum numbers as the photon. The lifetime of this fluctuation increases with theenergy of the parent photon Eγ and decreases with the square of the invariant mass of thepair M2

pair: ∆t ≈ 2Eγ/M2pair. When instead of a real photon one has a photon with virtuality

Q2 the fluctuation time is additionally suppressed to ∆t ≈ 2Eγ/(M2pair + Q2).

The main subject of this paper are interactions of high energy photons. It is useful tointroduce already in this place the commonly used terminology2. In the following we call thephoton direct if it interacts with another object as a whole quantity and we call the photonresolved if it interacts through one of the fermions produced in the quantum fluctuation.

If photon fluctuates into a pair of leptons, the process can be completely calculated withinQED. However, if it fluctuates into a pair of quarks, then the situation is much more compli-cated, because of QCD (quantum chromodynamics) interactions. The fact that photons canbehave as strongly interacting hadrons, is well known from soft, low energy γp interactions.The properties of those interactions are well described by the Vector Meson Dominance(VMD) model [3] – the photon turns first into a hadronic system with quantum numbers ofa vector meson and the hard interaction takes place between partons of the vector meson anda probing object. This contribution to the photon structure usually cannot be calculatedperturbatively and has to be parametrized in terms of the parton distribution functions inthe photon. Due to similarity to the structure of hadrons (e.g. proton) the contribution iscalled hadron-like. Only when the quark pair has a sufficient relative transverse momentum,the process is perturbatively calculable in QCD, and this contribution to the photon struc-ture we call point-like. Of course lepton pairs can give only the point-like contribution to thephoton structure, irrespectively of their relative transverse momentum. The contributionsto the structure of the photon discussed above are schematically shown in Fig.1, and thephoton wave function can be written as [4]:

1Throughout the paper we use the convention c = ~ = 1.2An attempt to sort out ambiguities existing in the terminology describing the hard interactions of photons

is performed in [2].

1

direct resolved

(a) (b) hadron-like (c) point-like

γ γ → V(JCP = 1−−) γ → f f

Figure 1: The direct photon (a) interacts as a whole quantity. The structure of the photonoriginates from quantum fluctuations – resolved photons: (b) hadron-like (c) point-like.

|γ〉 = cdirect|γdirect〉 +∑

V =ρ0,ω,...

cV |γV 〉 +∑

q=u,d,...

cq|γq〉 +∑

l=e,µ,τ

cl|γl〉 (1)

The coefficients cV , cq and cl depend on the factorization scale used to probe the photonand can be found e.g. in [5]. The coefficient cdirect is given by unitarity: c2

direct = 1−∑

c2V −

∑

c2q −

∑

c2l , and in practice is always close to unity. For the rich structure of the photon

developed through quantum fluctuations its investigation represents a fundamental test ofpredictions of both QED and QCD.

Most of the experimental results in the field of particle physics are today obtained fromexperiments performed at accelerators where beams of high energy elementary particlesare collided head on or with stationary targets. However, at present we do not have at ourdisposal photon beams and most of our knowledge on the structure of the photon comes frome+e− or ep colliders where the lepton beams serve as sources of high energy photons (theproton due to its large mass is a much weaker source of photons). The photons are emittedby the lepton beams in the bremsstrahlung process. Consequently the ‘photon beams’ haveno well defined energies, but rather are characterized by broad, continues spectra. The otherlimitation is that due to the four momentum conservation it is not possible for a beam leptonto emit a real photon in the bremsstrahlung process (although the distribution is peaked atvery small virtualities). Therefore in the present day experiments only the structure of quasi-real or virtual photons has been studied.

The main aim of this paper is to present and summarize the results on the structure ofthe photon and its interactions obtained from the LEP experiments. The classical way toinvestigate the structure of the photon at e+e− colliders is the measurement of the process:

e+(p1)e−(p2) → e+(p ′1)e

−(p ′2) X, (2)

proceeding via the interaction of two photons, which can be either quasi-real (γ) or virtual(γ⋆). The terms in parentheses in Eq. 2 represent the four-vectors of the particles as shownin Fig. 3a and X is a given leptonic or hadronic final state. The subdivision of the photonwave function presented in Eq. 1 corresponds to six main event classes in γγ → hadronsinteractions, characterized by some transverse momentum scale kT of the qq fluctuation andan unphysical scale k0 (of order 0.5 GeV) which divides the phase space into perturbative

2

a) kT1 = kT2 b) kT2 pTkT1 ) kT2 pTkT1 d) kT2 pTkT1 e)

kT2 pTkT1 f) kT2 pTkT1

Figure 2: The leading order diagrams corresponding to the six main event classes in theprocess γγ → hadrons: a) direct × direct; b) direct × point-like; c) point-like × point-like;d) direct × VMD; e) VMD × VMD; f) point-like × VMD.

and non-perturbative regions [4]. They are schematically shown in Fig. 2 and listed below(the quantity pT represents the maximum transverse momentum in the event):

• direct × direct – the photons directly produce a quark pair, (k0 ≪ kT1 = kT2),

• direct × point-like – the point-like photon splits into a qq pair and one of them (or adaughter thereof) interacts directly with the other photon, (k0 ≪ kT2 ≪ pT ≪ kT1),

• point-like × point-like – both photons perturbatively split into qq pairs, and subse-quently one parton from each photon takes part in hard interaction, (k0 ≪ kT1, kT2 ≪ pT ),

• direct × VMD – a direct photon interacts with the partons of VMD photon, (kT2 <k0 ≪ pT ≪ kT1),

• VMD × VMD – both photons turn first into hadrons and then interact like hadrons,(kT1, kT2 < k0, arbitrary pT ),

• point-like × VMD – the point-like photon perturbatively splits into a qq pair and andone of these (or a daughter parton thereof) interacts with a parton from VMD photon,(kT2 < k0 ≪ kT1 ≪ pT ),

Lepton pairs, in the above subdivision, can be produced only in the direct × direct process.

3

Depending on the virtualities of the photons involved in the process from Eq. 2 the scat-tered electrons3 may be observed in the detectors. From the experimental point of viewthe following three event classes are distinguished. In the case where none of the scatteredbeam electrons is observed in detector (anti-tagged), the structure of the quasi-real photonhas been studied at LEP in terms of total cross-sections, jet production, and heavy quarkproduction. If only one electron is observed (single-tagged), the process can be described asdeep-inelastic electron scattering off a quasi-real photon. These events have been studied tomeasure QED and QCD photon structure functions as well as QCD structure function ofthe electron. If both electrons are observed (double-tagged), the dynamics of highly virtualphoton collisions is probed. The QED and QCD structure of the interactions of two highlyvirtual photons has also been studied at LEP in terms of the effective structure function ofthe virtual photon and total cross sections.

In this paper we discuss only the results obtained in single tagged and double taggedmeasurements. The paper is organized as follows. In Section 2 the kinematics of two photoninteractions at e+e− colliders is presented and all kinematic quantities used later are defined.The cross section for the process e+e− → e+e− f f proceeding via the exchange of two photonsis discussed in Section 3 and its several important kinematical limits are derived. Section4 is devoted to the discussion of the deep inelastic electron-photon and electron-electronscattering processes. It starts with the introduction of the photon structure functions. Thenthe factorization of the e+e− cross section into fluxes of transverse and longitudinal photonsand the corresponding cross sections for electron photon scattering, expressed in terms ofthe photon structure functions, is presented. The QED and the hadronic structure functionsof the photon are discussed in detail, including the short description of the existing parame-terizations of the parton distribution functions in the photon. The structure functions of theelectron are defined and the relation between the photon and electron structure functions isgiven. The interactions of highly virtual photons are discussed in Section 5 in terms of theeffective structure function of the virtual photon and the total γ⋆γ⋆ cross sections, focusingon possible BFKL [36] effects in interactions of two highly virtual photons. The short pre-sentation of the Monte Carlo models used in experimental analyses of the photon structureis given in Section 5. In Section 6 the experimental aspects of the two photon physics at LEPare discussed on the example of the OPAL detector. The experimental results on the photonstructure obtained by the LEP experiments in single and double tagged measurements arepresented in Sections 8 and 9, respectively. When similar results from non LEP experimentsexist they are compared with the LEP measurements. The experimental results presented inthe paper include the QED and hadronic structure functions of both quasi-real and virtualphotons, the structure function of the electron and cross sections for the scattering of twovirtual photons with leptonic and hadronic final states. The numerical values of all exper-imental results presented in the paper are collected in the tables in four appendices. Thesummary of the studies of the photon and electron structure performed at LEP as well asthe prospects of the future measurements are given in Section 10.

There are many review articles which partly overlap or extend the material presentedin this paper. The survey in Ref. [6] consists of the short presentations of the most of ex-perimental results on the structure of the photon available at the end of the year 2000. In

3Electrons and positrons are generically referred to as electrons.

4

depth discussion of the photon structure functions and their measurements in deep inelasticelectron photon scattering process can be found in Ref. [7]. The thirty years old review byBudnev et al. in Ref. [8], although it lacks new experimental results, still serves as a source ofmany useful formulas and ideas. A very comprehensive discussions of the theoretical aspectsof two photon interactions and especially the approach based on structure functions is givenin Ref. [9] and Ref. [11]. The other interesting studies include Refs. [10, 12, 13].

The author of this thesis has been a member of the OPAL Collaboration since 1998. Iwas engaged primary in physics analyses of LEP2 data. I have concentrated on two photonphysics and I made essential contributions to the following OPAL publications:

• OPAL Collaboration, G. Abbiendi et al., Measurement of the hadronic cross section

for the scattering of two virtual photons at LEP, Eur.Phys.J. C24 (2002) 17.

• OPAL Collaboration, G. Abbiendi et al., Measurement of the photon and electron

structure functions in deep inelastic eγ and ee scattering at LEP, to be published inEur. Phys. J. C.

I have presented on behalf of the OPAL and other LEP collaborations several results on thefollowing conferences:

• Measurement of the Cross Section for the Process ee → eeγ⋆γ⋆ → eeX at√

see =189 GeV, International Conference on The Structure and Interactions of the Photon,Photon 1999, 23 - 27 May, Freiburg in Br., Germany.

• Measurement of the Cross Section for the Process ee → eeγ⋆γ⋆ → eeX at√

see =189 − 202 GeV, International Conference on The Structure and Interactions of thePhoton, Photon 2000, 26 - 31 August, Ambleside, UK.

• Measurement of the Hadronic Cross Section for the Scattering of Two Virtual Photons

at LEP, International Europhysics Conference on High Energy Physics EPS HEP 2001,12 - 18 July, 2001, Budapest, Hungary.

• Measurement of the Hadronic Cross Section for the Scattering of Two Virtual Photons

at OPAL, International Conference on The Structure and Interactions of the Photon,Photon 2001, 2 - 7 September, Ascona, Switzerland.

• Summary of the Photon Structure Function Measurements at LEP, 10th InternationalWorkshop on Deep Inelastic Scattering (DIS2002), 30 April - 4 May 2002, Krakow,Poland.

• Measurements of the Photon and Electron Structure Functions at LEP, XXXIII In-ternational Symposium on Multiparticle Dynamics, 5 - 11 September, 2003, Krakow,Poland.

5

During my stay at CERN as a postdoctoral fellow in the years 1998–2000 I participated inthe everyday running of the OPAL experiment. In particular I served as an online expert onshifts at the experiment. I was involved in the work of the luminosity group. I participated inthe estimation of the geometrical acceptance of the newly installed small angle ‘Far Forward’detector. As a member of the OPAL Two Photon Group I participated in the discussionson several two photon physics analyses and publications on this subject, especially thoseconnected with structure functions measurements.

6

2 Kinematics

The kinematics of two photon interactions at e+e− colliders is illustrated in Fig. 3a. Thequantities in parentheses represent the four vectors of the particles. The polar angles θi atwhich the electrons are scattered are measured with respect to the direction of original beamelectrons. Throughout the paper, unless said differently, i = 1, 2 denotes quantities whichare connected with the upper and lower vertex in Fig. 3a, respectively. The virtualities ofthe radiated photons are given by:

Q2i ≡ −q2

i = −(pi − p′

i)2 > 0. (3)

In case when the virtualities of the exchanged photons are significantly different, the processcan be interpreted as a deep inelastic scattering of the electron off the photon radiated bythe other beam electron (Fig. 3b) or directly off the other beam electron (Fig. 3c). The usualdimensionless variables of deep inelastic scattering are defined as:

yei =q1 · q2

pi · q3−i, xi =

Q2i

2q1 · q2, zi =

Q2i

2qi · p3−i, (4)

where, in infinite momentum of the target particle frame of reference, xi (zi) are fractions ofparton momentum with respect to the target photon (electron) and yei are the energies lostby the inelastically scattered electrons. The e+e− centre-of-mass energy squared is given bysee = (p1+p2)

2 and the hadronic (or leptonic, if a pair of leptons is produced in the final state)invariant mass (or the two-photon centre-of-mass energy) squared by W 2 ≡ sγγ = (q1 + q2)2.The e+e− and γγ centre-of-mass energies are related through: sγγ = ye1ye2see − Q2

1 − Q22.

The following useful relation between the above kinematical variables holds: seeziyei = Q2i .

Experimentally, the kinematical variables Q2i , yei, xi and zi are obtained from the four-

vectors of the tagged electrons and the hadronic final state via:

Q2i = 4EbE

′

i sin2(θi/2), (5)

yei = 1 − E ′i

Eb

cos2(θi/2), (6)

xi =Q2

i

Q21 + W 2 + Q2

2

, (7)

zi =Q2

i

yeisee=

E′

i sin2(θi/2)

Eb − E′

i cos2(θi/2). (8)

where Eb and E ′i refer to the energy of the beam electrons and the scattered electrons,

respectively, and the mass me of the electron has been neglected. If the beams collide headon and have equal energies Eb then see = 4E2

b. The two photon centre-of-mass energy,W , can be obtained from the energies, Eh, and momenta, ~ph, of final state particles (h),excluding the scattered electrons, via:

W 2 =

(

∑

h

Eh

)2

−(

∑

h

~ph

)2

. (9)

7

e(p1)(a) e(p01)�1 (?)(q1)e(p2) e(p02)�2 (?)(q2) �X e(k)(b) e(k0)� ?(q)

e(l) e(l0) (p) �Xe(k)( ) e(k0)� ?(q)e(l) e(l0) �X

Figure 3: The diagrams corresponding to the process e+e− → e+e− X, where X represents agiven leptonic or hadronic final state, in different versions discussed in the paper: a) generaldiagram, b) deep inelastic electron photon scattering, c) deep inelastic electron electronscattering.

In single tagged events, using conservation of energy and momentum, and assuming thatthe untagged electron travels along the beam direction one can calculate W using the for-mula [16]:

W 2 = (pb+ − ptag+)∑

h

ph− − p2tag,T (10)

where p± = E ± pz with pb+ and ptag+ being calculated for the tagged electron beforeand after scattering, respectively, and ptag,T being the transverse momentum of the taggedelectron. In double tagged events W can be determined from the four-momenta of the beamand the scattered electrons as follows:

W 2 =(

2Eb − E′

1 − E′

2

)2

−(

~p′

1 + ~p′

2

)2

(11)

For the special case when the virtualities of the exchanged photons are significantly different(deep inelastic scattering) we introduce the following notation (see Fig.2b,c):

Q2 ≡ −q2 = max (Q21, Q

22), P 2 ≡ −p2 = min (Q2

1, Q22) . (12)

? ? ee�� ? ff

e�?�

Figure 4: The scattering angles φ, θ⋆ and χ defined in the photon-photon centre of masssystem. For the precise definitions of the angles see text.

8

Then the kinematic variables x, z, ye refer to the photon with higher virtuality, and weintroduce the notation δ = P 2/2(p · q). The relation W 2 = Q2(1/x− 1) − P 2, which followsfrom Eq. 7, is often exploited in the following.

It is useful to define in this section the following additional angles (see Fig. 4). Theazimuthal angle φ is defined as the angle between the two scattering planes of the electronsin the photon-photon centre-of-mass system. The polar angle θ⋆ is defined as the anglebetween the produced fermion or antifermion and the photon-photon axis in the photon-photon centre-of-mass system. The azimuthal angle χ is defined as the angle between theelectron scattering plane and the scattering plane of the fermion which in the photon-photoncentre-of-mass system is scattered at cos θ⋆ < 0.

9

3 Cross sections for the process e+e− → e+e− f f

The four main diagrams representing the leading order contributions to the process e+e− →e+e− f f are shown in Fig. 5. In the kinematical region discussed in this paper – small ormedium virtualities Q2

i of at least one of the exchanged photons – the dominant contributionstems from the multiperipheral diagram [17]. The contributions from the other types ofdiagrams or from Z boson exchange become important only at very large photon virtualitiesand are not considered in the following.

The differential cross-section for the scattering of two unpolarized electrons via theexchange of two photons (multiperipheral diagram from Fig. 5a), integrated over the phasespace of produced particles ff, is given by (derivation of this formula can be found e.g.in [8, 9]):

d6σ =d3p

′1d

3p′2

E ′1 E ′

2

α2

16π4Q21Q

22

[

(q1 · q2)2 − Q2

1Q22

(p1 · p2)2 − m2em

2e

]1/2(

4ρ++1 ρ++

2 σTT + 2ρ++1 ρ00

2 σTL

+2ρ001 ρ++

2 σLT + ρ001 ρ00

2 σLL + 2|ρ+−1 ρ+−

2 |τTT cos 2φ − 8|ρ+01 ρ+0

2 |τTL cos φ)

(13)

where φ is the angle between the two scattering planes of the electrons in two photon centre-of-mass system (see Fig. 4a) and the four vectors and other kinematical variables are definedin Section 2. The quantities ρjk

1 and ρjk2 , where j, k ∈ (+,−, 0) denote the photon helicities,

are elements of the photon density matrix, which can be expressed in terms of the measurablemomenta pi and p′i (respectively qi) and are to that extent entirely known. They take thefollowing form [8]:

2ρ++i =

(2pi · q3−i − q1 · q2)2

(q1 · q2)2 − Q21Q

22

+ 1 − 4m2

e

Q2i

, ρ00i =

(2pi · q3−i − q1 · q2)2

(q1 · q2)2 − Q21Q

22

− 1 ,

|ρ+−i | = ρ++

i − 1 , |ρ+0i | =

√

(ρ00i + 1)|ρ+−

i | . (14)

The cross-sections σTT, σTL, σLT and σLL and the interference terms τTT and τTL correspondto specific helicity states of the interacting photons (T – transverse and L – longitudinal).e(a) e0 (?)

e e0 (?) f�fe(b) e0 (?)e e0

(?) f�fe( ) e0

(?)e e0 (?)f�fe(d) e0

(?)e e0 (?) f�f

Figure 5: Examples of the diagrams contributing in the leading order to the process e+e− →e+e− f f: a) multiperipheral, b) bremsstrahlung, c) annihilation, d) conversion. In each caseonly one possible diagram is shown.

10

They are obtained in QED from doubly virtual box diagram γ⋆(Q21)γ⋆(Q2

2) → f f and forthe case of lepton pair production in the leading order have the following form [8] (Slightmodifications with respect to the original formulas given in the reference [8] come from theintroduction of additional variables β and β defined in Eq. 17 and a change in the meaningof the variable T . These modifications are in accord with the modern notation used in theexpressions for the photon structre functions introduced in the next section.):

σTT =πα2

W 2 X

{

(q1 · q2)L

[

2 +2m2

X−(

2m2

q1 · q2

)2

− Q21 + Q2

2

X+

Q21 Q2

2 W 2

2X(q1 · q2)2

+3

4

(

Q21 Q2

2

X (q1 · q2)

)2]

− ∆t

[

1 +m2

X− Q2

1 + Q22

X+

1

T

Q21Q

22

(q1 · q2)2+

3

4

Q21 Q2

2

X2

]

}

σTL =πα2 Q2

2

W 2X2

{

∆t

[

1 − 1

T

Q21

(q1 · q2)2

(

6m2 − Q21 +

3

2

Q21 Q2

2

X

)]

− L

q1 · q2

[

4m2X − Q21(W 2 + 2m2) + Q2

1

(

Q21 + Q2

2 −3

2

Q21 Q2

2

X

)]}

σLT = σTL(Q21 ↔ Q2

2)

σLL =πα2 Q2

1 Q22

W 2X3

{

L

q1 · q2(2W 2X + 3Q2

1 Q22) − ∆t

(

2 +1

T

Q21 Q2

2

(q1 · q2)2

)}

τTT = − πα2

4W 2X

{

2∆t

X

[

2m2 +(Q2

1 − Q22)

2

W 2+

3

2

Q21 Q2

2

X

]

+L

q1 · q2

[

16m4

+16m2(Q21 + Q2

2) − 4Q21Q

22

(

2 +2m2

X− Q2

1 + Q22

X+

3

4

Q21Q

22

X2

)]}

τTL = −πα2√

Q21 Q2

2

W 2X2

{

L

(

2m2 − Q21 − Q2

2 +3

2

Q21 Q2

2

X

)

+ ∆t

(

2 − 3

2

q1 · q2

X

)}

(15)

where m is the lepton mass, and

X =(q1 · q2)2

W 2β2 , ∆t = 2(q1 · q2) ββ ,

T = 1 − β2β2 , L = ln1 + ββ

1 − ββ(16)

with

β =

√

1 − 4 m2

W 2, β =

√

1 − Q21 Q2

2

(q1 · q2)2(17)

Using the relation q1 ·q2 = 12(W 2+Q2

1+Q22), it is clear that the cross sections and interference

terms in Eq. 15 depend only on Q21, Q2

2, W 2 and the lepton mass m.It is worth of pointing out here that, in general, although the interference terms τTT and τTL

are independent of φ, even after integration of Eq. 13 over the full range of φ the dependenceon the interference terms does not vanish. This follows from strong kinematical correlations

11

of φ and the variables Q21, Q2

2 and W 2. In those regions of phase space where the interferenceterms are large there is no clear relation between the structure function approach discussedin the next section and the individual cross section terms. In such case the total or differen-tial cross sections are the most appropriate quantities to be measured by experiments.

The formulae given in Eq. 15 can be used in the case of quark pair production after multi-plying each cross section and interference term by Nce

4q , with Nc being the number of colours

and eq the quark charge, replacing the lepton mass m by a quark mass mq and summing overall active flavours. In fixed order perturbation theory the doubly virtual box contributionγ⋆(Q2

1)γ⋆(Q2

2) → qq is referred to as the quark parton model (QPM) approximation. In thecase of QED Eq. 15 can be applied irrespective of the virtualities Q2

1 and Q22, unless they

are small enough so that Z boson exchange can be neglected. The situation is different inQCD and depends on the relative sizes of the scales (Q2

1, Q22, W 2) characterizing the process

and a typical hadronic scale, which is of order of Λ. In QCD Λ denotes a scale at which theeffective coupling becomes large (see Eq. 50 and discussion after it). If the virtualities ofboth photons are large compared to Λ (Q2

1 , Q22 ≫ Λ2) we have a purely perturbative process

where the predictions of Eq. 15 are applicable. Furthermore, if in addition Q21 ≫ Q2

2, thanthe structure of the virtual target photon γ(Q2

2) is resolved by a probe photon γ(Q21). In

the high energy limit of photon–photon collisions (Q21 , Q2

2 ≪ W 2) one can directly studythe BFKL [36] dynamics. In that case Eq. 15 constitute a lowest order result. If one of thephotons is highly virtual and the other quasi–real (Q2

1 ≫ Λ2 & Q22) perturbation theory is

not reliable due to non-perturbative, long-distance effects. Therefore Eq. 15 are not directlyapplicable in this case, and the photon structure functions of the quasi–real photon arefactorized into non-perturbative parton distribution functions to be fixed by experimentalinformation and a calculable short distance coefficient functions as explained in the nextsection.

Below we discuss several useful kinematical limits of Eq. 15. They can be applied toboth lepton and quark (with appropriate normalization, see above) pairs in the final state.The formulas obtained below will be directly used in the next section to construct photonstructure functions in different approximations.

In the general Bjorken limit (m2 , Q22 ≪ Q2

1) we neglect expressions of order O(Q22/Q

21)

and O(m2/W 2). In this limit σLL and τTL vanish and the remaining quantities have thefollowing form:

σTT =2πα2

q1 · q2

{

L

(

1 − 1

2

Q21 W 2

(q1 · q2)2

)

− 1 +Q2

1 W 2

(q1 · q2)2− 1

T

Q21 Q2

2

(q1 · q2)2

}

σTL =2πα2

q1 · q2

Q22 W 2

(q1 · q2)4

Q41

T

σLT =2πα2

q1 · q2

Q21 W 2

(q1 · q2)2

τTT = − πα2

q1 · q2

Q41

(q1 · q2)2(18)

where

T =4m2

W 2+

Q21 Q2

2

(q1 · q2)2, L = ln

4

T. (19)

12

The above expressions further reduce in the limit of m2 = 0 (e.g. light quarks) and Q22 ≪ Q2

1:

σTT =2πα2

q1 · q2

{

L

(

1 − 1

2

Q21 W 2

(q1 · q2)2

)

− 2 +Q2

1 W 2

(q1 · q2)2

}

σTL = σLT =2πα2

q1 · q2

Q21 W 2

(q1 · q2)2

τTT = − πα2

q1 · q2

Q41

(q1 · q2)2(20)

where L = ln (4(q1 · q2)2/Q21 Q2

2). In the Bjorken limit the two-photon processes can beinterpreted as deep inelastic scattering and the appropriate cross sections can be describedin terms of virtual photon structure functions.

In the case where one photon is real (e.g. Q22 = 0), β = 1 and it follows from Eq. 15

that the terms σTL, σLL and τTL vanish and the remaining take the following form (the fullmass dependence is kept, which is relevant for the heavy quark contribution to the photonstructure functions):

σTT =πα2

(q1 · q2)3

{

L[

2(q1 · q2)2 + 2m2W 2 − 4m4 − Q21W

2]

− ∆t

q1 · q2

[

(q1 · q2)2 + m2W 2 − Q21W

2]

}

σLT =πα2Q2

1

(q1 · q2)4

[

W 2∆t − 4m2(q1 · q2)L]

τTT = − πα2

2(q1 · q2)4

[

∆t(2m2W 2 + Q41) + 8m2(q1 · q2)(m

2 + Q21)L]

(21)

where

∆t = (W 2 + Q21)β , L = ln

1 + β

1 − β. (22)

If in addition m2 ≪ Q21, the Eq. 21 reduce to:

σTT =πα2

(q1 · q2)

{[

2 − Q21 W 2

(q1 · q2)2

]

lnW 2

m2− W 2 + Q2

1

q1 · q2

[

1 − Q21 W 2

(q1 · q2)2

]}

σLT =πα2Q2

1

(q1 · q2)4

[

W 2(W 2 + Q21)]

τTT = − πα2

2(q1 · q2)4

[

Q41(W

2 + Q21)]

(23)

Near the mass shell (Q2i → 0) the terms of longitudinal photon scattering vanish. In this

limit σTT and τTT are transformed into the corresponding quantities for real two photonprocesses. In particular, at Q2

i = 0, σTT coincides with the cross section σγγ of the γγ → Xtransition for real non-polarized photons. As a result, at Q2

i → 0, we get:

σTT(W 2, Q21, Q

22) → σγγ(W 2) =

4πα2

W 2

[(

1 +4m2

W 2− 8m4

W 4

)

L −(

1

W 2+

4m2

W 4

)

∆t

]

,

τTT(W 2, Q21, Q

22) → τγγ(W 2) = −16πα2m2

W 6(∆t + 2m2L) , (24)

13

where ∆t and L are given by Eq. 22 with Q21 set to zero. The other cross sections vanish in

the limit Q2i → 0 as:

σTL ∼ Q22 , σLT ∼ Q2

1 , σLL ∼ Q21Q

22 , τTL ∼

√

Q21Q

22 . (25)

The full cross sections and their several limits discussed in this Section will be later used toconstruct photon structure functions in various approximations.

14

4 Deep inelastic eγ and ee scattering

In this section we assume that the virtuality of one of the exchanged photons is much smallerthan the virtuality of the other photon (P 2 ≪ Q2). In that case the subprocess eγ → effcan be interpreted as deep inelastic electron–photon scattering (see Fig. 3b) in which thestructure of the quasi–real (P 2 . Λ2) or virtual (P 2 & Λ2) transverse photon is probed by thevirtual photon of both transverse and longitudinal polarizations. Due to experimental (andkinematical) limitations the virtuality of the quasi–real target photon can be kept small butit is always larger than zero. The effect of this small virtuality is usually neglected althoughit might be important. This problem can be overcome when one interprets the processas deep inelastic scattering of an electron off other, target electron (see Fig. 3c). In suchsituation the virtual photon probes directly the structure of the real electron. Although thefirst interpretation (DIS eγ) is most widely used, in the following we discuss both approaches.

4.1 Photon structure functions

In the case of lepton pair production, the cross section presented in the previous section inEq. 13 is determined by QED and contain full information needed to describe the reaction.On the other hand in the case of quark pair production, QCD corrections are involved andit becomes necessary to parameterize the cross section by means of the photon structurefunctions. However, to have consistent description of both QED and hadronic structure ofthe photon we usually express the cross sections for both lepton and quark pair production interms of structure functions. Usually one introduces structure functions for a spin–averagedtarget photon. They can be expressed in terms of the photon–photon cross sections σab

(a, b = T, L) defined in the previous section as follows [8, 9]:

F γ2 (x, Q2, P 2) =

Q2

4π2α

1

β

[

σTT(x, Q2, P 2) + σLT(x, Q2, P 2)

−1

2σLL(x, Q2, P 2) − 1

2σTL(x, Q2, P 2)

]

2xF γT (x, Q2, P 2) =

Q2

4π2αβ

[

σTT(x, Q2, P 2) − 1

2σTL(x, Q2, P 2)

]

F γL(x, Q2, P 2) =

Q2

4π2αβ

[

σLT(x, Q2, P 2) − 1

2σLL(x, Q2, P 2)

]

(26)

where β =√

1 − Q2 P 2/(p · q)2 and we made use of Eq. 7 to express the cross sections σab asfunctions of x, Q2 and P 2. This choice of variables is conveniently used for the presentationof the photon structure functions. The following relation holds between the above structurefunctions: F γ

L = β2 F γ2 −2xF γ

T . The expressions in Eq. 26 are generally valid for arbitrary P 2

and Q2, although they have a meaningful interpretation as structure functions of a targetphoton probed by deeply virtual photon only in the limit P 2 ≪ Q2. Since the fluxes oftransverse and longitudinal photons are different it is useful to introduce structure functions

15

of transverse (a = T ) and longitudinal (a = L) target photons:

F γa

2 (x, Q2, P 2) =Q2

4π2α

1

β

[

σTa(x, Q2, P 2) + σLa(x, Q2, P 2)]

2xF γa

T (x, Q2, P 2) =Q2

4π2αβ σTa(x, Q2, P 2)

F γa

L (x, Q2, P 2) =Q2

4π2αβ σLa(x, Q2, P 2) (27)

They are related to each other via F γa

L = β2 F γa

2 −2xF γa

T . The expressions for a spin-averagedtarget photons given in Eq. 26 are related to the structure functions of polarized photonsgiven in Eq. 27 via F γ

i = F γT

i − 12F γL

i , where i = 2, L, T .

It is well known that for P 2 = 0 the cross section for the process ee → eeγγ⋆ → eeX canbe factorised into a product of the flux of target photons of transverse polarization and thecross section for deep inelastic eγ scattering [7]. One can also show [9] that in the Bjorkenlimit (Q2 → ∞, p · q → ∞, x – fixed, what means that the quantity δ = P 2/2p · q = xP 2/Q2

is small and can be neglected) the factorization holds for virtual target photons as well.Neglecting the terms of order O(

√δ), changing appropriately the variables and integrating

over azimuthal angles of the scattered electrons one can rewrite the cross section given inEq. 13 in the factorised form [9]:

dσ(ee → eeX)

dxdQ2dydP 2= fT

γ⋆/e(y, P 2)dσ(eγT → eX)

dxdQ2+ fL

γ⋆/e(y, P 2)dσ(eγL → eX)

dxdQ2(28)

where y is the fractional momentum of the target photon with respect to the beam elec-tron. The fluxes of transverse and longitudinal target photons in the equivalent photonapproximation (EPA) are given by [18]:

fTγ⋆/e(y, P 2) =

α

2π

[

1 + (1 − y)2

y

1

P 2− 2y

m2e

P 4

]

(29)

fLγ⋆/e(y, P 2) =

α

2π

2(1 − y)

y

1

P 2(30)

The cross sections for deep inelastic electron–photon scattering expressed in terms of thepolarized photon structure functions F γa

2 and F γa

L read:

dσ(eγa → eX)

dxdQ2=

2πα2

xQ4

[(

1 + (1 − ye)2)

F γa

2 (x, Q2, P 2) − y2eF

γa

L (x, Q2, P 2)]

(31)

In the limit of small virtualities of target photons (P 2 ≈ 0) the contribution from thelongitudinal target photons vanish and the cross section for the process ee → eeγγ⋆ → eeXcan be written as a product of the flux of transversally polarized photons and the crosssection for deep inelastic electron photon scattering:

d4σ

dxdQ2dzdP 2=

2πα2

x2Q4

[(

1 + (1 − ye)2)

F γ2 (x, Q2, P 2) − y2

eFγL(x, Q2, P 2)

]

fTγ⋆/e(z/x, P 2) (32)

16

y

f^ T γ /e (

y,P

2 ) [G

eV-2

],

fT γ /

e (y,

P 2

max

)

f^ Tγ/e (y,P2) f

^ Tγ/e (y,P 2ave )

f Tγ/e (y,P 2max )

P 2min

P 2max (y)

P 2ave P 2max (y) P 2ave (y)

Eb θmax P 2max (y=0) P 2ave

LEP1 45.6 GeV 34 mrad 2.4 GeV2 0.07 GeV2

LEP2 100. GeV 34 mrad 11.6 GeV2 0.30 GeV2

10-4

10-2

1

10 2

10 4

10 6

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 6: Comparison of EPA (fTγ⋆/e) and the Weizsacker–Williams approximation (fT

γ⋆/e).

The chosen beam energies, minimum tagging angle, the values of P 2max(y = 0) and the values

of P 2ave averaged over the full range of P 2 and the range of 10−4 < y < 1 are also shown.

The flux of equivalent photons as well as the structure functions depend in principle on thevirtuality of the target photon. However, in experiments the virtuality of the target photon isusually unknown and one has to calculate structure functions assuming some effective targetvirtuality, P 2

eff , and integrate the flux over the range of possible virtualities. In practice, thiseffective virtuality P 2

eff is usually obtained from Monte Carlo simulation or from the best fitof the structure function predictions to the data. The integration of EPA over the possiblerange of P 2 leads to the Weizsacker–Williams approximation [19]:

fTγ⋆/e(y, P 2

max) =

∫ P 2max

P 2

min(y)

dP 2fTγ⋆/e(y, P 2) =

=α

2π

[

1 + (1 − y)2

yln

(

P 2max(1 − y)

m2ey

2

)

− 21 − y

y+ 2y

m2e

P 2max

]

(33)

where

P 2min(y) =

m2ey

2

1 − yand P 2

max(y) = (1 − y)E2θ2max (34)

The lower boundary of the integration P 2min follows from the four vector conservation and

the upper boundary is determined by the experimental acceptance. In practical applications

17

we often choose a constant value of Pmax (equal to the minimum virtuality of probe photons)and fill the missing phase space by a model prediction. In Fig. 6 we show the predictions ofEPA and the Weizsacker-Williams approximation for two typical situations during LEP1 andLEP2 data taking periods. The chosen beam energies are 45 GeV and 100 GeV, respectively,and in both cases the minimum tagging angle had been chosen at 34 mrad. Predictions ofEPA are shown at P 2

min, P 2max(y) and at average values of virtuality P 2

ave(y) depending on yand at the virtuality P 2

ave averaged over y in the range 10−4 < y < 1. The predictions ofthe Weizsacker-Williams approximation is shown for the integration range given by Eqs. 34.For the LEP1 case the Weizsacker–Williams approximation is close to the EPA predictionat P 2

ave. But this is already not the case for the LEP2 energies, and follows from only a veryweak dependence of the Weizsacker–Williams approximation on the upper boundary of theP 2 integration range.

Let us return to the discussion of the cross section of Eq. 32. For small values of the vari-able ye, accessible at LEP energies, the term proportional to F γ

L can be neglected. IntegratingEq. 32 over P 2 we get:

d3σ

dxdQ2dz≈ 2πα2

x2Q4

(

1 + (1 − ye)2)

F γ2 (x, Q2, P 2

eff)fγ⋆/e(z/x, P 2max) (35)

where only a weak dependence of F γ2 on P 2 has been assumed. The effective value of P 2

eff

has been chosen in such a way that the following relation is fulfilled:

〈F γ2 (P 2)〉 = F γ

2 (P 2eff) (36)

One can integrate Eq. 35 over z changing the variables according to ye = Q2/(zsee) ≡ χ/zand using the relation:

1

x

∫ x

zmin

dz(

1 + (1 − χ/z)2)

fγ⋆/e

(

z/x, P 2max

)

=

=

∫ 1

η

dy(

1 + (1 − η/y)2) fγ⋆/e(y, P 2max) ≡ K(η, b) (37)

where zmin = χ, η ≡ ymin = zmin/x = χ/x and b = P 2max/m

2e. The function K(η, b) has the

following form:

K(η, b) =α

2π

[

−1

6π2(η + 2)2 − ln (η)(η + 2)2 ln

(

b

η

)

+ (10 − 4η − 3η2) ln (η)

− 1

2(3η + 19)(η − 1) + 2(η + 3)(η − 1) ln (b(1 − η)) + (η + 2)2Li2(η)

]

(38)

where Li2(x) = −∫ x

0ln |1−t|

tdt is the dilogarithm function. Usually we assume that P 2

eff = 0and in consequence the formula given in Eq. 35 can be rewritten in the form:

d2σee

dxdQ2≈ 2πα2

xQ4F γ

2 (x, Q2)K(

Q2

xs,P 2

max

m2e

)

(39)

which can be directly used to obtain the photon structure function F γ2 from the measured

differential cross section d2σee/dx dQ2.

18

4.2 QED structure functions of the photon

The QED structure functions of the photon can be defined for any type of lepton pairs in thefinal state of the process eγ → e ll. However, measurements exist only for muon pairs andtherefore we will use the muon mass in the numerical results presented in this paragraph.

The leading order formulae for the QED structure functions F γ2,QED and F γ

L,QED keepingthe full dependence on the virtuality of the quasi-real photon P 2 can be obtained fromEq. 26 together with the cross sections listed in Eq. 15. They are however very long andwill be not written down here. Much more compact formulae can be derived in the limit2xP 2/Q2 ≪ 1, but keeping the terms of order O(P 2/m2). They are referred to as theBethe-Heitler expressions for the virtual photon and have the following form [20]:

F γ2,BH(x, Q2, P 2) =

α

πx

{

[x2 + (1 − x)2] ln1 + ββ

1 − ββ− β + 6βx(1 − x)

+

[

2x(1 − x) − 1 − β2

1 − β2− (1 − β2)(1 − x)2

]

ββ(1 − β2)

1 − β2β2

+(1 − β2)(1 − x)

[

1

2(1 − x)(1 + β2) − 2x

]

ln1 + ββ

1 − ββ

}

(40)

F γL,BH(x, Q2, P 2) =

α

πx2(1 − x)

[

β − 1

2(1 − β2) ln

1 + ββ

1 − ββ

]

(41)

where β and β are given in Eq. 17. Neglecting in addition the terms of order O(m2/W 2) wearrive at the Bjorken limit discussed in Section 3. The structure functions in this limit canbe constructed using Eq. 26 and the cross sections listed in Eq. 18. A very compact formulafor F γ

2,QED is obtained in the Bjorken limit combined with the restriction P 2 ≪ W 2:

F γ2,apr(x, Q2, P 2) =

α

πx

{

[

x2 + (1 − x)2]

lnW 2

m2 + P 2x(1 − x)− 1

+8x(1 − x) +P 2(1 − x)

m2 + P 2x(1 − x)

}

(42)

The predictions of the exact formula for F γ2,QED and of the approximations discussed above

are shown in Fig. 7 for different values of P 2 = 0, 0.01, 0.1 and 1 GeV2 and a moderate valueof Q2 = 10 GeV2. Both, exact and approximate F γ

2,QED are strongly suppressed with P 2.They agree with each other for small values of P 2, but start to differ for P 2 & 0.1 GeV2. Theformula from Eq. 42 gives already rather bad approximation in the whole x range, whereasthe other two approximations underestimate the exact result only in the large x region. Thisends the discussion of the QED structure functions of the virtual photon and the rest of theparagraph is devoted to the QED structure functions of the real photon.

The differential cross-section given in Eq. 13 can be expressed in terms which havethe same angular dependence with respect to the azimuthal angles χ and φ (see Fig. 4for the definition) and combinations thereof, using 13 structure functions as shown in [85].In the case of the final state of the process e+e− → e+e− f f being a pair of leptons it isexperimentally possible to measure a triple differential cross section. By integrating over

19

x

Fγ 2,

QE

D /α

Fγ2, QED(x,Q2,P2)

Fγ2, BH(x,Q2,P2)

Fγ2, apr(x,Q2,P2)

Fγ2, Bj(x,Q2,P2)

Q2 = 10 GeV2 P2 = 0 GeV2 P2 =0.01 GeV2

P2 =0.1 GeV2

P2 = 1 GeV2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

Fγ 2,

QE

D /α

Fγ2,QED(x,Q2)

Fγ2,QED(x,Q2), β=1

Q2 = 1 GeV2

Q2 = 30 GeV2

x

Fγ A

,QE

D /α

FγA,QED(x,Q2)

FγA,QED(x,Q2), β=1

Q2 = 1 GeV2

Q2 = 30 GeV2

x

Fγ B

,QE

D /α

FγB,QED(x,Q2)

FγL,QED(x,Q2)

FγB,QED(x,Q2) = F

γL,QED(x,Q2), β=1

Q2 = 1 GeV2

Q2 = 30 GeV2

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.3

-0.2

-0.1

0

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 7: The structure function F γ2,QED for µ+µ− final states based on double virtual box

calculation (solid) and its approximations discussed in the text: the Bethe-Heitler expression(dense dots), the Bjorken limit (rare dots), the formula given in Eq. 42 (dash). The curvesare plotted for Q2 = 10 GeV2 and for several values of P 2 = 0, 0.01, 0.1, 1 GeV2.

Figure 8: The structure functions of the real photon F γ2,QED, F γ

A,QED, F γB,QED and F γ

L,QED forµ+µ− final states at Q2 = 1 GeV2 and 30 GeV2 with full mass dependence (solid and dots)and in the leading logarithmic approximation discussed in the text (dash).

all angular dependences except the χ dependence, the cross section for deep inelastic eγscattering can be written as:

d3σ(eγ →eff)

dxdQ2dχ/2π=

2πα2

xQ4

(

1 + (1 − ye)2)

[

F γ2,QED − (1 − ǫ(ye))F

γL,QED

−ρ(ye)FγA,QED cos χ +

1

2ǫ(ye)F

γB,QED cos 2χ

]

(43)

where the functions ρ(ye) = (2−ye)√

1 − ye/(1+(1−ye)2) and ǫ(ye) = 2(1−ye)/(1+(1−ye)

2)are both close to unity for small values of ye accessed at LEP. The above cross section is basedon the structure functions of the real photon, P 2 = 0. The structure functions F γ

2,QED andF γ

L,QED can be obtained from the Eq. 26 and the cross sections given in Eq. 21. The structurefunctions F γ

A,QED and F γB,QED are new. They are proportional to the cross sections for the

target photon to interact with different polarization states of the virtual photon: transverse–longitudinal interference (A) and interference between the two transverse polarizations (B).The formulae for the structure functions F γ

A,QED, F γB,QED, F γ

L,QED and F γ2,QED, which keep the

full dependence on the lepton mass up to terms of order O(m2/W 2), taken from [84] read:

F γA,QED(x, Q2) =

4α

πx√

x(1 − x)(1 − 2x)

{

β

[

1 + (1 − β2)1 − x

1 − 2x

]

20

+3x − 2

1 − 2x

√

1 − β2 arccos(

√

1 − β2)

}

F γB,QED(x, Q2) =

4α

πx2(1 − x)

{

β

[

1 − (1 − β2)1 − x

2x

]

+1

2(1 − β2)

[

1 − 2x

x− 1 − x

2x(1 − β2)

]

ln1 + β

1 − β

}

F γL,QED(x, Q2) =

α

πx2(1 − x)

[

β − 1

2(1 − β2) ln

1 + β

1 − β

]

F γ2,QED(x, Q2) =

α

πx

{

[

x2 + (1 − x)2]

ln1 + β

1 − β− β + 8βx(1 − x) − β(1 − β2)(1 − x)2

+(1 − β2)(1 − x) ·[

1

2(1 − x)(1 + β2) − 2x

]

ln1 + β

1 − β

}

(44)

where β =√

1 − 4m2

W 2 =√

1 − 4m2

Q2

x1−x

. These are in fact the familiar massive Bethe-Heitler

expressions for the real photon. The structure functions in the leading logarithmic approxi-mation can be obtained from Eq. 44 in the limit β → 1. They have the following form:

F γA,QED(x, Q2) =

4α

πx√

x(1 − x)(1 − 2x)

F γB,QED(x, Q2) = F γ

L,QED(x, Q2) =4α

πx2(1 − x)

F γ2,QED(x, Q2) =

α

πx

{

[

x2 + (1 − x)2]

lnW 2

m2− 1 + 8x(1 − x)

}

(45)

The formulae for F γ2,QED and F γ

L,QED in the leading logarithmic approximation are related tothe cross sections listed in Section 3 through the Eq. 26 and the cross sections given in Eq. 23.In the above approximation only F γ

2,QED has a non-trivial dependence on Q2. It must benoted, that although in the above leading logarithmic approximation the structure functionsF γ

B,QED and F γL,QED are accidentally described by the same function of x, they involve different

photon helicity structures. In F γB,QED the photons are purely transverse. The comparison of

the structure functions F γA,QED, F γ

B,QED, F γ2,QED and F γ

L,QED obtained with mass dependentterms (Eq. 44) and in the leading logarithmic approximation (Eq. 45) is shown in Fig. 8 fortwo values of Q2 = 1 GeV2 and 30 GeV2. The approximation is rather good in the case ofF γ

2,QED, but for the structure functions F γA,QED and F γ

B,QED the difference between the exactand approximate formulas is significant especially in the high x region and for low values ofQ2. One can also see that the structure functions F γ

B,QED and F γL,QED are very close to each

other in the whole x and Q2 ranges considered here, also when the mass dependent termsare taken into account. Therefore the measurement of F γ

B,QED gives a good estimate of thelongitudinal structure function F γ

L,QED which is not measured by experiments.

4.3 Hadronic structure functions of the photon

The hadronic structure of the photon follows from the quantum fluctuation of the photoninto a pair of quarks. In general both the real and the virtual photon possess a parton

21

content. One expects that the parton distributions of the virtual photon in the limit ofP 2 = 0 smoothly transform into the parton distributions of the real photon. Formallythe treatments of the real and the virtual photons are similar, although they differ in theboundary conditions used for the calculation of the parton distribution functions. In thefollowing we discuss the more general case, the virtual photon, pointing out the differenceswith respect to the real photon. The discussion is concentrated on transverse (real andvirtual) photons and only a short notice on the longitudinal photons will be given at theend of this paragraph. That means we require that P 2 ≪ Q2 to guarantee that the physicalcross sections are dominated by the transverse target photon contributions. The discussionpresented in this paragraph is mainly based on the following publications [23, 27, 74, 101].

The predictions for the structure functions F γ2 and F γ

L in the quark parton model (QPM)approximation can be obtained from the formulae for the QED structure functions discussedin the previous paragraph by formally multiplying them by Nc

∑nf

k=1 e4qk

, where Nc is thenumber of colours, eqk

is the electric charge of the flavour qk and the sum runs over all activeflavours nf . As an example let us rewrite here the leading logarithmic results, which in caseof QED were given in Eq. 45. They take the following form:

F γ2,QPM(x, Q2) =

Nc α

π

nf∑

k=1

e4qk

x

{

[

x2 + (1 − x)2]

lnW 2

m2qk

− 1 + 8x(1 − x)

}

F γL,QPM(x, Q2) =

4Nc α

π

nf∑

k=1

e4qk

x2(1 − x) (46)

The above QPM result for F γ2 is valid for the real photon. In case of the virtual photon, its

virtuality P 2 can act as the regulator and no quark masses have to be introduced. Takingthe limit P 2/Q2 → 0 in Eq. 40 whenever possible we obtain [21, 22]:

F γ2,QPM(x, Q2, P 2) =

Nc α

π

nf∑

k=1

e4qk

x

{

[

x2 + (1 − x)2]

lnQ2

x2 P 2− 2 + 6x(1 − x)

}

(47)

The structure function F γL,QPM does not need a regulator and remains unchanged. It should

be noted that the virtual photon structure functions are kinematically constrained within0 ≤ x ≤ (1 + P 2/Q2)−1 [22]. It is also worth of pointing out here that the QPM resultsalready predict a logarithmic evolution of the photon structure function F γ

2 with Q2. Thisis in contrast to the behaviour of the proton (or hadron in general) structure function F p

2 ,in case of which the scaling violations appear only due to the QCD corrections. The reasonof this different behaviour is the point-like coupling of the photon to quarks. It leads tothe rise of F γ

2 towards large values of x, where the proton structure function F p2 decreases.

This point-like coupling also results in the positive scaling violation of F γ2 for all values of x,

whereas F p2 exhibits positive scaling violations at small values of x (caused by quark pairs

production from gluons) and negative scaling violations at large values of x (caused by gluonradiation). In case of the photon, the direct production of quark pairs at large values of x ismore efficient than the loss of quarks in that region due to gluon radiation.

Due to the QCD corrections the leading order QED results given in Eq. 46 and Eq. 47are not sufficient. For the purpose of the following discussion it is useful to decompose the

22

hadronic photon structure functions into parts corresponding to light F γi,l and heavy F γ

i,h

quarks:F γ

i (x, Q2, P 2) = F γi,l(x, Q2, P 2) + F γ

i,h(x, Q2, P 2) (48)

where i = 2, L. In terms of the parton distribution functions in the photon, the structurefunctions for the light quarks are given in next-to-leading order (NLO) QCD in the modifiedminimal subtraction factorization scheme (MS) by (see e.g. [27]):

F γi,l(x, Q2, P 2) = 2x

3∑

k=1

e2qk

{

qγk (x, Q2, P 2) +

α

2πe2

qkCi,γ(x)+

αs(Q2)

2π

∫ 1

x

dy

y

(

Ci,q

(

x

y

)

qγk (y, Q2, P 2) + Ci,g

(

x

y

)

gγ(y, Q2, P 2)

)}

(49)

where the quark qγk(x, Q2, P 2) and gluon gγ(x, Q2, P 2) parton densities provide the hadron-

like contributions of the photon to F γ2 , while Ci,γ provides the point-like contribution. Note

that due to charge conjugation invariance qγk(x, Q2, P 2) = qγ

k (x, Q2, P 2). The running strongcoupling constant αs(Q

2) in NLO is given by:

αs(Q2)

4π≃ 1

β0 ln Q2/Λ2− β1

β30

ln ln Q2/Λ2

(ln Q2/Λ2)2(50)

with β0 = 11− 2nf/3 , β1 = 102− 38nf/3. For Q2 values much larger than Λ2, the effectivecoupling is small and a perturbative description in terms of quarks and gluons interactingweakly makes sense. For Q2 of order of Λ2 we can not make such a picture, since quarksand gluons will arrange themselves into strongly bound clusters (hadrons). The value of Λis not predicted by the theory: it is a free parameter to be determined from experiment. Itis of order of typical hadronic mass with a value somewhere in the range 0.1 – 0.5 GeV. TheWilson coefficient functions Ci,q, Ci,g and Ci,γ are given by [28, 29, 101]:

C2,q(x) =4

3

[

1 + x2

(1 − x)+

(

ln1 − x

x− 3

4

)

+1

4(9 + 5x)

]

, CL,q(x) =8

3x , (51)

C2,g(x) =1

2

{

[x2 + (1 − x)2] ln1 − x

x− 1 + 8x(1 − x)

}

, CL,g(x) = 2x(1 − x) , (52)

CMS2,γ (x) =

3

1/2C2,g(x) , CMS

L,γ (x) =3

1/2CL,g(x) . (53)

The regularized function (1 − x)−1+ in C2,q is defined by the so called plus prescription:

∫ 1

0

dxf(x)

(1 − x)+

≡∫ 1

0

dxf(x) − f(1)

(1 − x)(54)

In next-to-leading order there exist a freedom in the definition of terms belonging to theparton density functions and the terms which are included in the hard scattering matrixelements. The different choices, called factorization schemes, make that physics quantities,like F γ

2 , calculated in fixed order perturbation theory can differ by finite terms (see e.g. [95]).These ambiguities disappear only when the calculations are performed to all orders. There

23

are two commonly used factorization schemes for the photon structure function: the MSscheme and the DISγ scheme. In the original DIS scheme, introduced for the proton, allhigher order corrections have been absorbed into the definition of the quark distributionfunctions, so that F p

2 was proportional in all orders in αs to the quark distribution functions.In case of photon, in the DISγ scheme only the C2,γ term has been absorbed into the NLO(MS) quark densities, so that:

qγk,DISγ

= qγ

k,MS+

α

2πe2

qkCMS

2,γ , CDISγ

2,γ = 0 (55)

gγDISγ

= gγ

MS(56)

Note that CL,γ is the same in the MS and DISγ schemes. The LO expressions for F γi can

be obtained from Eq. 49 dropping all higher order terms Cq,g,γ and the term proportional toβ1 in Eq. 50.

The above discussion applies only to the light quarks u, d, s. Due to the large scaleintroduced by the heavy quark masses, their contribution to the photon structure functionhave to be calculated in fixed order perturbation theory and can be expressed as a sum of apoint-like (pl) and a hadron-like (hl) parts:

F γi,h(x, Q2, P 2) = F γ, pl

i,h (x, Q2, P 2) + F γ, hli,h (x, Q2, P 2) (57)

This is schematically shown in Fig. 9. The point-like contribution can be approximated bythe Bethe-Heitler formula (the lowest order QED result for doubly virtual box diagram)given in Eq. 41 multiplied by Nc e4

q. The relevant expressions for F γ, pl2,h and F γ, pl

L,h read:

F γ, pl2,h (x, Q2, P 2) = Nc

e4q α

πx

{

[x2 + (1 − x)2] ln1 + ββ

1 − ββ− β + 6βx(1 − x)

+

[

2x(1 − x) − 1 − β2

1 − β2− (1 − β2)(1 − x)2

]

ββ(1 − β2)

1 − β2β2

+(1 − β2)(1 − x)

[

1

2(1 − x)(1 + β2) − 2x

]

ln1 + ββ

1 − ββ

}

(58)

F γ, plL,h (x, Q2, P 2) = Nc

e4q α

πx2(1 − x)

[

β − 1

2(1 − β2) ln

(

1 + ββ

1 − ββ

)]

(59)

e(a) e0 ?e e0 Q�Q

e(b) e0 ?e e0 Q�Q

Figure 9: Leading order diagrams of the point-like (a) and hadron-like (b) contribution tothe heavy quark structure function of the photon F γ

2,h.

24

In principle one should use here the NLO formulas. However the NLO expressions have beenonly calculated so far for the real photon and they were found to make only a small correctioncomparable to the ambiguities due to different choices of the mass of heavy flavours [26].The resolved heavy quark contribution to the photon structure functions is calculated via

the process γ⋆g → QQ shown in Fig. 9b with f γ⋆g→QQi (x, Q2, P 2) given by Eq. 58 or Eq. 59

and with e2qα replaced by e2

qαs(µ2F )/6:

F γ, hli,h (x, Q2, P 2) =

∫ zmax

zmin

dz

zzgγ(z, µ2

F , P 2)fγ⋆g→QQi

(x

z, Q2, P 2

)

(60)

where zmin = x(1 + 4m2Q/Q2 + P 2/Q2) and zmax = (1 + P 2/Q2)−1. Here gγ represents the

gluon distribution function in the appropriate order, and the factorization scale µ2F can be

chosen as µ2F ≃ 4m2

h [24] or µ2F = Q2 + 4m2

h [25] (in the later case the relation µ2F ≫ P 2 is

fulfilled also for large P 2).

The quark, gluon and photon distribution functions in the transverse photon, denoted byqγk , gγ and Γγ, respectively, obey the following inhomogeneous Dokshitzer-Gribov-Lipatov-

Altarelli-Parisi (DGLAP) [51] evolution equations for the massless parton densities:

∂qγi

∂ ln Q2= Pqγ ⊗ Γγ + 2

nf∑

k=1

Pqq ⊗ qγk + Pqg ⊗ gγ

∂gγ

∂ ln Q2= Pgγ ⊗ Γγ + 2

nf∑

k=1

Pgq ⊗ qγk + Pgg ⊗ gγ

∂Γγ

∂ ln Q2= Pγγ ⊗ Γγ + 2

nf∑

k=1

Pγq ⊗ qγk + Pγg ⊗ gγ (61)

where the explicit dependence of the distribution functions on x, Q2 and P 2 has been omitted.The symbol ⊗ stands for the convolution integral:

(f ⊗ g)(x) =

∫ 1

x

dy

yf

(

x

y

)

g(y) . (62)

The Altarelli-Parisi evolution kernels Pij are generalized splitting functions:

Pij(y, α, αs) =

∞∑

l,m=0

αlαms

(2π)l+mP

(l,m)ij (y) . (63)

Since the electromagnetic coupling constant is small we can neglect terms of order O(α2) aswell as the dependence of α on Q2 in the evolution equations given in Eq. 61. The partondistribution functions qγ and gγ are already of order O(α) and therefore we can set l = 0 inall generalized evolution kernels Pij which are multiplied by these parton densities. In caseof Pqγ, Pgγ and Pγγ the only contributions of order O(α) come from setting l = 1 in Eq. 63.The terms proportional to Pγq and Pγg are necessarily of order O(α2) and can be dropped.

25

q qgPqq; Pgq g qqPqg g ggPgg qqPq Figure 10: Diagrams illustrating the DGLAP splitting functions in the leading order.

In consequence the evolution equation for the photon distribution Γγ decouples and can besolved separately (see e.g. [23]). The remaining evolution equation can be written as:

∂qγk

∂ ln Q2=

α

2πPqγ +

αs(Q2)

2π

[

Pqq ⊗ qγk + Pqg ⊗ gγ

]

,

∂gγ

∂ ln Q2=

α

2πPgγ +

αs(Q2)

2π

[ nf∑

k=1

Pgq ⊗ qγk + Pgg ⊗ gγ

]

. (64)

The hadronic splitting functions P (x, Q2) receive the following LO and NLO QCD contri-butions:

P (y, Q2) = P (0)(y) +αs(Q

2)

2πP (1)(y) (65)

In the LO QCD the following parton splittings occur: q → qg, q → gq, g → qq, g → gg.They are schematically shown in Fig. 10 and are described by the following formulae [30,31]:

P (0)qγ (y) = Nc e4

qk[y2 + (1 − y)2] , P (0)

qiqk(y) = δik

[

4

3

1 + y2

(1 − y)++ 2δ(1 − y)

]

,

P (0)qg (y) =

1

2

[

y2 + (1 − y)2]

, P (0)gq (y) =

4

3

1 + (1 − y)2

y, P (0)

gγ (y) = 0 ,

P (0)gg (y) = 6

[

1 − y

y+

y

(1 − y)++ y(1 − y) +

(

11

12− nf

18

)

δ(1 − y)

]

. (66)

The expressions for the splitting functions in NLO can be found e.g. in [31, 32].For completeness let us make a short notice on the longitudinal target photon. For-

mally one can apply all the considerations of this paragraph to longitudinal photons settingΓγL(x, Q2, P 2) = 0 [33]. This condition follows from the the fact that to order O(α) thereis no transverse photon inside the longitudinal target photon. In consequence the quarkand gluon distribution functions of the longitudinal photon satisfy homogeneous evolutionequations [33].

4.4 Parton distribution functions in the photon

There are many parameterizations of parton distribution functions in the photon availabletoday, constructed in the leading and/or next-to-leading order. Most of them are valid onlyfor the real photon, but few have been also constructed for the virtual photon (P 2 & Λ2).

26

Below we discuss shortly the main features of the most popular parameterizations. If notsaid explicitly a parameterization is valid for the real photon and is constructed in DISγ

factorization scheme. The names of the parameterizations are usually constructed from thefirst letters of the names of their authors. For more details the reader is referred to theoriginal publications or to the summary given e.g. in [7].

1. DG – Drees and Grassie [92]: This is the oldest parameterization of parton distri-butions in the photon. The parameterization is based on the solution of the leading orderevolution equations. The x dependent input parton distributions, with free parameters, wereassumed at Q2

0 = 1 GeV2 and fitted to the only data available at that time from PLUTOat Q2

0 = 5.3 GeV2 [122]. Due to limited statistics further assumptions have been made:qγd = qγ

s , qγu = qγ

c , and the input gluon distribution function has been set to zero, whichmeans that gluons are generated purely dynamically. The charm and bottom quarks aretreated as massless, and are included only for Q2 > 20 GeV2 and Q2 > 200 GeV2, respec-tively, by means of the number of flavours nf used in the evolution equations. The user canchoose from three independent sets constructed for nf = 3, 4, 5.

2. LAC – Levy, Abramowicz, Charchu la [93]: The parameterisation is based on thesolution of the leading order evolution equations. The input parton distributions of the form:

xq0(x, Q20) = Ae2

qxx2 + (1 − x)2

1 − B ln (1 − x)+ CxD(1 − x)E

xg0(x, Q20) = Cgx

Dg(1 − x)Eg (67)

were assumed at Q20 = 4 GeV2 (sets LAC1 and LAC2) or at Q2

0 = 1 GeV2 (set LAC3) andwere fitted to the available data. The sets LAC1 and LAC2 differ in the gluon distributionfunction – in LAC2 the parameter Dg is set to zero. The first and second terms in the quarkdistribution function correspond to the point-like and hadron-like parts, respectively. Thecharm quark is treated as massless and its contribution is only included for W 2 > 4m2

c . Noparton distribution for the bottom quark is available.

3. WHIT – Watanabe, Hagiwara, Izubuchi, Tanaka [94]: The parametrisation isbased on the solution of the leading order evolution equations. The initial parton distribu-tions for three light flavours are assumed at Q2

0 = 4 GeV2. The contribution from charmquarks (mc = 1.5 GeV) is added according to the Bethe–Heitler formula for Q2 < 100 GeV2

or by using the massive–quark evolution equations for Q2 > 100 GeV2. No parton distribu-tion function for bottom quarks is available. The parametrisation is essentially a study of thesensitivity of the photon structure function to the gluon content of the photon. The initialgluon distribution is parametrized by the simple formula xg0(x, Q2

0)/α = Ag(Cg +1)(1−x)Cg .The available data are not accurate enough to determine the gluon parameters Ag and Cg.Therefore the authors provide six parton distributions which have systematically differentgluon contents: WHIT1-3 (Ag = 0.5) and WHIT4-6 (Ag = 1) and in both cases Cg = 3, 9, 15.

4. GRV – Gluck, Reya, Vogt [74]: The parton distribution functions are available inleading order and in next-to-leading order. They are evolved from low starting scales (Q2

0 =0.25 GeV2 in LO and Q2

0 = 0.3 GeV2 in NLO) using a VMD input based on measurement

27

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

10-3

10-2

10-1

LAC1LAC2LAC3

Q2=100 GeV2

Q2=5 GeV2

x

F2γ /α

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

10-3

10-2

10-1

WHIT1WHIT2WHIT3

WHIT4WHIT5WHIT6

Q2=100 GeV2

Q2=5 GeV2

x

F2γ /α

0

0.2

0.4

0.6

0.8

1

1.2

10-3

10-2

10-1

SaS1D

SaS1M

SaS2D

SaS2M

Q2=100 GeV2

Q2=5 GeV2

x

F2γ /α

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

10-3

10-2

10-1

CJKL

Q2=1, 5, 15, 100, 1000 GeV2

x

F2γ /α

Figure 11: The photon structure function F γ2 divided by the fine structure constant obtained

from different parton distribution functions. Shown are predictions from different sets ofLAC, WHIT and SaS parameterizations at two values of Q2 = 5, 100 GeV2 and from theCJKL parameterization for Q2 = 1, 5, 15, 100, 1000 GeV2.

of the pion structure function of the form q(x, Q20) = g(x, Q2

0) = κ(4πα/f 2ρ )fπ(x, Q2

0 wherexfπ ∼ xa(1 − x)b and 1/f 2

ρ = 2.2. The similarity of the ρ and π mesons is assumed anda proportionality factor 1 ≤ κ ≤ 2 is used to account for the inclusion of ω, φ and otherhigh mass vector mesons. The point-like contribution is chosen to vanish at the input scaleand for higher virtualities is generated dynamically using the full evolution equations. Thecharm and bottom quarks are included via the Bethe-Heitler formula with mc = 1.5 GeVand mb = 4.5 GeV.

5. AFG – Aurenche, Fontannaz, Guillet [95]: The parametrisation is available only inthe next-to-leading order. Similarly as in the GRV parametrisation the point-like contribu-tion vanishes at the low starting scale chosen at Q2

0 = 0.5 GeV2 and the purely hadron-likeinput is based on VMD arguments, where a coherent sum of low mass vector mesons ρ,ω and φ is used. The evolution is performed in the massless scheme for three flavours forQ2 < m2

c = 2 GeV2 and for four flavours for Q2 > m2c . No parton distribution function

for bottom quarks is available. The AFG parton distributions are constructed in the MSfactorization scheme.

6. CJKL – Cornet, Jankowski, Krawczyk, Lorca [96]: This is the most recent pa-rameterization. A global, three parameter fit to all data available today is performed basedon the leading order evolution equations. The idea of radiatively generated parton distri-butions similar to the one used in GRV parameterization is exploited. The starting scaleis chosen at Q2

0 = 0.25 GeV2. However, input densities of the ρ0 meson are not approx-imated by the pionic ones, but instead the valence-like and gluon densities of the formxvρ(x, Q2

0) = Nvxα(1 − x)β and xg(x, Q2

0) = Ngxα(1 − x)β are used. All sea quark distri-

28

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10-3

10-2

10-1

GRV LOGRV HOGRSc LOGRSc HO

Q2=5, 100 GeV2

x

F2γ /α

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

GRV HOAFG HOGRSc HO

Q2=1, 15, 100 GeV2

x

F2γ /α

Figure 12: The photon structure function F γ2 divided by the fine structure constant obtained

from different parton distribution functions. Predictions of LO and HO GRV and GRScparameterization for two values of Q2 = 5, 100 GeV2 are shown on the left and the comparisonof HO predictions from GRV, GRSc and AFG parameterizations at Q2 = 1, 15, 100 GeV2 isshown on the right.

butions are neglected at the input scale. The special treatment of heavy quarks based onthe ACOT (χ) prescription [97], originally introduced for the proton structure function isadopted to the photon structure function for the first time. It means essentially an improve-ment of the treatment of the threshold region W ≈ 2mh where usually the Bethe-Heitlerformula is used.

7. GRS – Gluck, Reya, Stratmann [101]: The parameterization is an extension of thephenomenologically successful GRV photon densities [74] to non-zero P 2 in LO and NLO.Similarly as in GRV, at low input scale Q2

0 ≈ 0.25 GeV2, the parton densities of real photonsare given by a VMD inspired input. A simple prescription which smoothly interpolatesbetween P 2 = 0 and P 2 ≫ Λ2 is used:

fγ(P 2)(x, Q2 = P 2) = η(P 2)fγ(P 2)non−pert(x, P 2) + [1 − η(P 2)]f

γ(P 2)pert (x, P 2) (68)

with P 2 = max(P 2, µ2) and η(P 2) = (1 − P 2/m2ρ)−2 where mρ is some effective mass in the

vector meson propagator. The VMD-like non-perturbative input is taken to be proportional

to the GRV pion densities fγ(P 2)non−pert(x, P 2) = κ(4πα/f 2

ρ )fπ(x, P 2). Heavy quarks do not takepart in the Q2 evolution and they must be taken into account using Bethe-Heitler formula.For real photons it is known up to NLO and can be found in [26]. For non-zero P 2 they areavailable only in LO and the relevant cross sections are given in Eq. 15.

8. GRSc – Gluck, Reya, Schienbein [27]: The parton distribution functions are con-structed for the real photon in LO and NLO and for the virtual photon in LO, which within

29

0

0.1

0.2

0.3

0.4

0.5

0.6

10-3

10-2

10-1

SaS1D P2 = 0 GeV2 Q2=15 GeV2

P2 = 0.01 GeV2

P2 = 0.1 GeV2

P2 = 1 GeV2

x

F2γ /α

0

0.1

0.2

0.3

0.4

0.5

0.6

10-3

10-2

10-1

GRSc LO P2 = 0 GeV2 Q2=15 GeV2

P2 = 0.01 GeV2

P2 = 0.1 GeV2

P2 = 1 GeV2

x

F2γ /α

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

SaS1DQ2=2,75,1000 GeV2

x=0.03 GRSc LO

GRS LO

P2 [GeV2]

F2γ /α

Figure 13: The photon structure function F γ2 divided by the fine structure constant of the

virtual photon. The predictions of SaS1D and GRSc parameterizations for Q2 = 15 GeV2

and for different virtualities of the target photons P 2 = 0, 0, 01, 0.1, 1 GeV2.

Figure 14: The P 2 dependence of F γ2 /α predicted by SaS1D, GRSc and GRS parameteriza-

tions at a medium x = 0.03 and three values of Q2 = 2, 75, 1000 GeV2.

sufficient accuracy may be also used in NLO. A consistent set of boundary conditions havebeen formulated, which allow for a calculation of f γ(x, Q2, P 2) also in NLO QCD as well asfor a smooth transition from the virtual to the real photon. The parameterization is basedon the same general ideas as the GRV parameterization. However, in contrast to GRV, thecoherent sum of vector mesons is used in VMD to describe the hadronic components of thephoton, with the recently updated parton distribution of the pion [101]. The charm quarkmass has been changed to mc = 1.4 GeV.In case of virtual photons it is assumed that the effects of the photon virtuality are entirelytaken into account by the flux factors, which are valid for Q2 ≫ P 2. In consequence, incontrast to GRS parameterization, all partonic cross sections are calculated as if P 2 = 0. Inparticular, the process γγ⋆ → qq is used to evaluate Cγ(x), instead of the process γ⋆γ⋆ → qqand charm contribution for P 2 > 0 is based on the Bethe-Heitler formula for the real photon.

9. SaS – Schuler, Sjostrand [102,103]: The parameterization is based on the solution ofthe leading order evolution equations for three massless flavours. Two sets were constructedwith different starting scales: Q2

0 = 0.36 GeV2 (SaS1) and Q20 = 4 GeV2 (SaS2). Each set is

available in the MS (set M) and DISγ (set D) factorization schemes. The set SaS2 containsa larger VMD contribution in comparison to the set SaS1, which is necessary in order to fitthe data due to a much larger starting scale Q2

0.The extension to non-zero P 2 is based on a dispersion-integral in the mass of qq fluctuations,which links perturbative and non-perturbative contributions [102, 103]. The qq fluctuations

30

are separated into a discrete sum of vector meson states and a high mass continuous per-turbative spectrum from the point-like contribution. Both terms are suppressed by differentP 2-dependent terms.The contributions from the charm (mc = 1.3 GeV) and bottom (mb = 4.6 GeV) quarks aretaken into account via the Bethe–Heitler formula taking into account full P 2 dependence.

The parameterizations of the parton distribution functions discussed above are used toconstruct the photon structure function F γ

2 . They are compared below for several values ofthe probe and target photon virtualities.

In Fig. 11 the structure function F γ2 for the real photon constructed from the parton

distribution functions predicted by different sets of LAC, WHIT and SaS parameterizationsare shown for two values of Q2 = 5 and 100 GeV2, and by the CJKL parameterization forQ2 = 1, 5, 15, 100, 1000 GeV2. The predictions of LAC and WHIT parameterizations arevery similar at large values of x. However, they start to differ significantly at low values ofx where the value of the starting scale Q2

0 and the shape of the input distribution of gluonsbecome important.

The structure functions F γ2 for the real photon constructed from parton distribution

functions predicted by the leading order (LO) and next-to-leading order (HO) GRV andGRSc parameterizations are shown in Fig. 12 for two values of Q2 = 5, 100 GeV2. At thelower value of Q2 the LO and HO GRV predictions are rather different in whole x range. Incase of GRSc the LO and HO predictions are very similar for x < 0.1 and differ only at largevalues of x. With increasing Q2 the LO and HO predictions of both parameterizations tendto be very similar at low values of x but they still differ significantly at high values of x. InFig. 12 we also show the comparison of the predictions for F γ

2 for the real photon obtainedfrom all available next-to-leading order parameterizations GRV, GRSc and AFG for threevalues of Q2 = 1, 15 and 100 GeV2. The predictions are quite different for large values of xwhere the data are not very precise.

In Fig14 we show the structure function F γ2 for the virtual photon obtained from the

parton distribution functions predicted by SaS, GRSc and GRS parameterizations. The xdependence of SaS1D and GRSc parameterizations at Q2 = 15 GeV2 and for several valuesof P 2 = 0, 0.01, 0.1, 1 GeV2 are shown on the left. Clearly the suppression of F γ

2 withincreasing P 2 and especially the hadron-like contribution are visible. On the right, the P 2

dependence of F γ2 predicted by SaS1D, GRSc and GRS parameterizations for three values

of Q2 = 2, 75, 1000 GeV2 and at moderately small value of x = 0.03 are shown. Once morea strong suppression of F γ

2 with increasing P 2 is visible.

4.5 Hadronic structure of the electron

At present beam energies the splitting of the electron induced reactions into the flux of quasi-real transverse photons in the electron and the distribution of partons inside the photon isquite a good approximation. However, at future colliders, contributions from W and Zbosons as well as from γ-Z interference will play more and more significant role. Thereforeinstead of measuring separate structure functions of each of intermediate bosons (which

31

0

2

4

6

8

10

12

14

16

18

20

10-3

10-2

10-1

z

F2e (z

) / α

2

Q2 = 15 GeV2 LAC1

CJKL

GRSc LO

GRV LO

WHIT1

SaS1D

QPM

z

F2e (z

) / α

2

Q2 = 5 GeV2

Q2 = 20 GeV2

GRSc LO

F2γ(x,Q2,P2=0)

F2γ(x,Q2,P2)

SaS1D LO

0

2

4

6

8

10

12

14

16

10-3

10-2

10-1

Figure 15: The electron structure function F e2 divided by the fine structure constant squared

obtained from different parton distribution functions in the photon using the Eq. 70. In allparameterizations of F γ

2 used, the P 2 dependence is either neglected or set to P 2 = 0.

Figure 16: The electron structure function F e2 divided by the fine structure constant squared

obtained from the parton distribution functions in the photon predicted by SaS1D and GRScLO parameterizations in two cases: P 2 set to zero in F γ

2 and with full P 2 dependence of F γ2

and of the photon flux and for two different values of Q2 = 5, 20 GeV2.

from experimental point of view might be even not possible) it seems more reasonable tomeasure the electron structure functions. The concept of the hadronic structure of theelectron has been introduced only recently in Refs. [104,105], where the evolution equationsin Q2 have been constructed and the asymptotic solutions have been found for the quarkand gluon content of the electron in the leading logarithmic approximation. At LEP energiesthe measurements of the electron structure function and the photon structure function areroughly equivalent. The cross section for the deep inelastic scattering of an electron off atarget electron expressed in terms of the electron structure functions F e

2 and F eL reads:

d2σ

dz dQ2=

2πα2

z Q4

[(

1 + (1 − ye)2)

F e2 (z, Q2) − y2

eFeL(z, Q2)

]

(69)

Neglecting the term proportional to F eL we can directly use this formula to extract the

electron structure function F e2 from the differential cross section. Comparing Eq. 69 with

Eq. 32 one finds the following relation between the electron F ea and the photon F γ

a (a = 2, L)structure functions:

F ea (z, Q2; P 2

max) ≡∫ 1

z

dx

∫ P 2max

P 2

min(z/x)

dP 2 z

x2F γ

a (x, Q2, P 2)fγ⋆/e(z/x, P 2) (70)

From the theoretical point of view, the measurement of the electron structure function, evenat present energies, has some advantages [106,107]. The target photons used to measure the

32

photon structure function, are always off-shell. Treating the photons as real is an approxima-tion. From the measurements of the QED structure function of the photon we know that theprecise information about this small but finite virtuality of the target photons is essential ina correct description of the data by QED predictions. In the case of hadronic structure func-tion of the photon the dependence on the virtuality of target photons in not measured norcompletely known theoretically. The photon structure function also depends on the formof the equivalent photon formula and this is still being discussed in literature [147]. Theabove problems do not appear in the concept of the electron structure function which onlyweakly depends on P 2

max. The most important drawback of the electron structure functionis that its shape is dominated by the strongly peaked flux of target photons. In Fig. 15 wepresent a comparison of the electron structure functions F e

2 constructed from different pa-rameterizations of the photon structure function F γ

2 using the formula given in Eq. 70, as afunction of z and for Q2 = 15 GeV2. When calculating the curves in Fig. 15, the dependenceof F γ

2 on P 2 has been neglected (P 2 = 0), which means that the virtuality of the photonradiated by the target electron enters only through the photon flux (Eq. 29) and throughthe upper limit of integration P 2

max = 4.5 GeV2 (typical value at LEP2). One can see thatthe shape of F e

2 is dominated by the strongly peaked EPA formula. The sensitivity of theelectron structure function F e

2 to the dependence of the photon structure function F γ2 on P 2

has been investigated in Fig. 16. The predictions based on GRSc LO [27] and SaS1D [102]parameterizations are shown for two different situations: with the full dependence of F γ

2

on P 2 and with P 2 set to zero in F γ2 . One can see that the difference between these two

situations is of the same order as the difference between the parameterizations themself.

33

5 Structure and interactions of virtual photons

In the previous Section we discussed in detail the interactions of virtual photons with quasi-real or slightly virtual photons. The cross section for this process has been expressed in termsof the structure functions of the quasi-real photon. In the present Section we shall extendour discussion to the interactions of two highly virtual photons (Q2, P 2 ≫ Λ2). From theexperimental point of view this means that we will consider double-tagged events, i.e. eventswith both final state electrons scattered at sufficiently large polar angles θi to be observedin a detector.

5.1 The cross section

The differential cross-section given in Eq. 13, in the limit Q2i ≫ m2

e, reduces to:

d6σ =d3p

′1d3p

′2

E ′1 E ′

2

LTT

(

σTT + ǫ1σLT + ǫ2σTL + ǫ1ǫ2σLL +1

2ǫ1ǫ2τTT cos 2φ

−√

2ǫ1(ǫ1 + 1)√

2ǫ2(ǫ2 + 1)τTL cos φ)

(71)

where ǫi ≡ ρ00i /2ρ++

i and the photon helicities ρ00i and ρ++

i are given in Eq. 14. The valuesof ǫi are close to unity in the kinematic region accessed at LEP (low values of yei). Theluminosity function LTT describes the flux of the incoming transversely polarized photons.Comparing Eq. 71 with the general formula for the cross section given in Eq. 13 we find thatLTT is given by:

LTT =α2

16π4Q21Q

22

[

(q1 · q2)2 − Q21Q

22

(p1 · p2)2 − m2em

2e

]1/2

4ρ++1 ρ++

2 (72)

The sum of the cross sections and interference terms given in parentheses in Eq. 71 representsthe cross section for the reaction γ⋆γ⋆ → hadrons (leptons):

σγ⋆γ⋆ = σTT + σTL + σLT + σLL +1

2τTT cos 2φ − 4τTL cos φ (73)

where we made use of the approximation ǫi ≈ 1. The cross section for the process e+e− →e+e− hadrons can be then written as a product of a term describing the flux of the incomingphotons and the cross-section for the interaction of the virtual photons. Depending on therelative size of virtualities of the interacting photons the process can be divided into twoclasses. When the virtualities of the photons differ significantly, Q2 ≫ P 2 ≫ Λ2, the processcan be considered as a deep inelastic scattering of the electron off the virtual photon. In thiscase the considerations of the previous Section apply and the cross section can be expressedin terms of virtual photon structure functions defined in analogy to the structure functionsof the quasi-real photon. However, experimentally one can access only an effective structurefunction F γ

eff = (Q2/4π2α)β−1 σγ⋆γ⋆ which is proportional to the cross section σγ⋆γ⋆ definedin Eq. 73. Making further assumptions, that σLL is negligible, σTL = σLT, and that the

34

x

(a)⟨Q2⟩ = 120 GeV2

⟨P2⟩ = 0.01 GeV2

Fγeff / α

F2γ / α

(F2γ+3/2β

– -2FLγT) / α

(F2γ+3/2FL

γ) / α

x

(b)⟨Q2⟩ = 120 GeV2

⟨P2⟩ = 3.70 GeV2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 17: Comparison of the effective structure function F γeff with F γ

2 and the approxi-mations given in Eq. 74 for two cases: (a) 〈Q2〉 = 120 GeV2, 〈P 2〉 = 0.01 GeV2 and (b)〈Q2〉 = 120 GeV2, 〈P 2〉 = 3.7 GeV2 assuming the QPM (mu = md = ms = 0.325 GeV andmc = 1.5 GeV). To a good approximation the quantity measured for the virtual photon isthe effective structure function given by (F γ

eff = F γ2 + 3/2F γ

L)/α.

interference terms do not contribute, one can relate F γeff to the known structure functions F γ

2

and F γL :

F γeff = F γ

2 +3

2β−2F γT

L ≃ F γ2 +

3

2F γ

L (74)

In Fig. 17 we compare the effective structure function F γeff, calculated in QPM, directly from

the cross section σγ⋆γ⋆ with neglected interference terms and using the approximations givenin Eq. 74. One can see that for the quasi-real photon the effective structure function F γ

eff isequal to F γ

2 and is well in agreement with the first relation given in Eq. 74. However, thesecond approximation in Eq. 74 is significantly worse. On the other hand in case of virtualphoton the effective structure function F γ

eff is close to the combination of F γ2 and F γ

L givenin Eq. 74, but very different from F γ

2 .In the case when the virtualities of the interacting photons are of the same order, the

usual interpretation in terms of the photon structure functions is not meaningful and themost appropriate quantity which can be measured is just the cross section.

35

5.2 DGLAP and/or BFKL dynamics?

The second topic extensively studied both theoretically and experimentally is the dynamicsof interactions of highly virtual photons. The interaction of two virtual photons has beenargued to be a ‘golden’ process to study the parton dynamics, because it does not require (forsufficiently large photon virtualities) any non-perturbative input in theoretical calculation.The discussion in this paragraph is based on the following publications [37–39]. Manyprocesses in QCD can be described by a fixed order expansion in the strong coupling constantαs. However, hadronic processes at high-energy colliders often involve more than one energyscale, and in consequence in perturbative QCD αs gets multiplied by a large logarithm ofthe relevant scales. In such a case fixed order calculations must be replaced with leading-logcalculations in which such terms are resumed.In Fig. 18 we show schematically the ‘map’ of deep inelastic electron photon scattering in(x, Q2) plane. If one of the colliding hadronic objects is much smaller than the other one(in our case the photons with higher and lower virtuality, respectively) then in order tocorrectly treat such collision it is necessary to resum terms proportional to (αs ln Q2/Q2

0)n,

because the smallness of αs is compensated by the large size of ln Q2/Q20. This procedure

leads to DGLAP [51] evolution equations introduced in Section 4. DGLAP evolution, upin Q2, describes the change of parton density with increasing spatial resolution of about1/Q (see Fig. 18). The evolution is self-contained in a given x interval and in particulardoes not involve knowledge of the parton densities at lower values of x. The cross section isproportional to the quark distribution at scale Q2, which is related to the qark distributionat another scale Q2

0 via:

xq(x, Q2) = xq(x, Q20) + αs ln

Q2

Q20

∫ 1

x

dy1x

y1Pqq

(

x

y1

)

q(y1, Q20)

+ αs lnQ2

Q20

∫ 1

x

dy1x

y1Pqg

(

x

y1

)

g(y1, Q20) + ... (75)

where the splitting functions Pqq and Pqg are given in the leading order by Eq. 66. In anappropriate gauge Eq. 75 can be interpreted as the first (with one rung only) in a series ofladder diagrams shown in Fig. 19, whose rungs are strongly ordered in transverse momenta:

Q21 ≫ k2

T,n ≫ k2T,n−1 ≫ · · · ≫ k2

T,1 ≫ Q22 (76)

For high energy scattering, i.e. at small x, the ladder becomes long, and because the domi-nation of the splitting function Pgg(y) ∼ 1/y in this region, it consists mainly of gluons. Thedominant terms in perturbative QCD expansion are of the form (αs ln (1/x) ln (Q2/Q2

0))n.

Their resummation is referred to as the double leading logarithmic approximation (DLLA).The region of its applicability is shown in Fig. 18. In this case in addition to the strongordering of the transverse momenta (see Eq. 76), also the fractional longitudinal momenta4

x′i where i = 1, ..., n, of the gluons along the ladder are strongly ordered:

x1 ≡ x′n ≪ x′

n−1 ≪ · · · ≪ x′1 ≡ x2 (77)

4In Eq. 77 and in Fig. 19 we use x′

ito denote the fractional longitudinal momenta of partons along the

ladder to avoid a conflict in notation with the symbols x1 and x2 used throughout the paper to denote theBjorken scaling variables of DIS electron photon scattering process.

36

����

������

������������

����

��������

��������

��������

���� ��

��������������

������

����

������

log

(1/x

)

����������

Region of highparton density

DGLAP eq.

BFK

L e

q.DLLA eq

.

Critical lin

e

log(Q )2

e(p1) e(p01) ?(q1)Q21 qx0n k2T;n �qx0n�1 k2T;n�1....e(p2) e(p02) ?(q2)Q22 �qx01 k2T;1 qx02 k2T;2

Figure 18: The (x, Q2) regions of validity of the various evolution equations. In DGLAPevolution with increasing Q2 the partonic constituents of the photon are resolved more finely.The BFKL evolution towards small values of x allows to access regions of growing gluondensities, but the resolution in transverse plane remains at approximately 1/Q. Simultaneousevolution in x and Q2 leads to DLLA evolution equations. Perturbative QCD is applicablebelow the critical line.

Figure 19: The ladder diagram corresponding to the interaction of two virtual photons. Theladders start and end with quark boxes, and can have gluon or quark rungs. The fractionallongitudinal momenta and the transverse momenta squared of each rung are denoted as x′

i

and kT,i, respectively, with i = 1, ..., n.

The DLLA leads to a gluon density strongly rising towards low x, like xg(x, Q2) ∼ x−λ withan effective power λ ≈

√

(12αs/π) ln (1/x) ln (Q2/Q20) [40].

When the partons at the ends of the ladder have similar transverse momenta, Q21 ≈ Q2

2,there is no phase space left for the transverse momenta to be strongly ordered along theladder, and the strong ordering in kT is replaced by a diffusion pattern. In perturbativeexpansion all terms proportional to (αs ln (1/x))n ≈ (αs ln (W 2/Q2))n need to be resumed.This is done by the Balitski-Fadin-Kuraev-Lipatov (BFKL) equation [36]. In the appropriategauge each term corresponds to the n-rung ladder diagram (Fig. 19) in which gluon emissionsare strongly ordered in longitudinal momenta. The basic quantity in this approximation isthe unintegrated (over the gluon transverse momentum) gluon density distribution G(x, k2

T )related to the usual gluon distribution g(x, Q2) via:

G(x, k2T ) = x

∂g(x, Q2)

∂ ln Q2

∣

∣

∣

∣

Q2=k2

T

(78)

37

= F0 + + + � � �Figure 20: The contributions to the γ⋆γ⋆ cross section: box digram (F0), one gluon exchange,next in the set of ladder diagrams, ... .

The unintegrated gluon distribution G(x, Q2) satisfies the following leading logarithmic(ln 1/x) BFKL evolution equation [36]:

∂G(x, k2T )

∂ ln 1/x= 3

αs(k2T )

πk2

T

∫ ∞

kT0

dk′2T

k′2T

[

G(x, k′2T ) − G(x, k2

T )

|k′2T − k2

T |+

G(x, k2T )

√

4k′4T + k4

T

]

(79)

where kT0 is the lower limit cut-off, which is necessary if the running coupling constanteffects are taken into account. The BFKL evolution equation takes us to smaller values of xwhere the gluon density increases while the transverse size that is probed remains the same.Eventually we reach the critical line where recombination effects must be incorporated viaperturbative QCD. Finally we enter the region of high parton density where we can not usenormal methods of perturbation theory. At higher Q2 we can evolve further in x beforereaching the critical line.

When the running of αs is neglected and k2T0 = 0 the BFKL equation can be solved

analytically giving at small x the following dependence of the gluon distribution:

xg(x, Q2) ∼ x1−αIP

√

ln (1/x)(80)

where αIP = 1 + λ = 1 + (3αs/π)4 ln 2 is the bare QCD pomeron. The value λ is usuallysaid to be 0.5, though this requires αs = 0.18, which happens only at high Q2. When therunning coupling constant effects are taken into account the small x behaviour of partondistributions is xg(x, Q2) ∼ x−λ, with λ ≈ 0.5 as in the fixed coupling case.

Recently, much attention has been given to the BFKL pomeron especially for small xdeep inelastic electron proton scattering at HERA. The onset of BFKL effects has beensearched for in the measurements of the proton structure function F p

2 and the analysis ofhadronic final states [47]. The situation is so far inconclusive, with the strongest hint forBFKL effects in the data coming from the analysis of forward π0 production.

It has been argued [45, 48–50] that the e+e− colliders offer an excellent opportunity totest the BFKL predictions through the measurement of the total cross section for the pro-cess γ⋆γ⋆ → hadrons. The cross section σγ⋆γ⋆ can be expressed in terms od ladder diagramssquared as shown in Fig. 20. The diagram denoted as F0 represents the lowest order con-tribution called the ‘box diagram’ or QPM contribution. The contribution of the next inline diagram is often called ‘one gluon exchange’ contribution. For sufficiently large photonvirtualities Q2

1 and Q22 (more than few GeV2), this BFKL calculations can be carried out

38

without non-perturbative input. If additionally Q21 ≃ Q2

2 then the evolution in Q2 is sup-pressed, allowing for a clean test of BFKL effects.

The original LO-BFKL calculations [45] predicted an increase of σγ⋆γ⋆ by a factor 20or more compared to calculations without BFKL effects, or with only DGLAP (Q2) evo-lution [51]. Since then the LO-BFKL calculations have been improved by including charmquark mass effects, running of the strong coupling constant αs and the contribution of thelongitudinal photon polarization states. Recently, it has become clear that the Next-to-Leading Order (NLO) corrections to the BFKL equation are large and effectively reduce thevalue of λ. A phenomenological determination of the Higher Order (HO) effects was pre-sented in Ref. [50] and the resulting BFKL scattering cross-sections were shown to increaseby a factor 2-3 only, relative to the calculations without BFKL effects. Since then theoret-ically motivated improved higher order calculations have been performed [38, 42, 43, 52–55],and give similar results.

For the comparison of the data to BFKL calculations the following additional kinematicquantity, which is a measure of the length of the gluon ladder, is defined [45]:

Y ≡ ln

(

seeye1ye2√

Q21Q

22

)

= ln

(

W 2 + Q21 + Q2

2√

Q21Q

22

)

≃ ln

(

W 2

√

Q21Q

22

)

≡ Y , (81)

where the approximation requires W 2 ≫ Q2i .

39

6 Monte Carlo generators

In the analysis of the data collected by experiments at high energy colliders a crucial role isplayed by Monte Carlo generators. They are based on our knowledge of the physics processestaking place during a collision of elementary particles. The usual output from a Monte Carloprogram is a collection of four momenta of particles produced in the interaction. At thisstage, often called hadron (or more generically generator) level, we have an exact informationon the whole final state. However, in the real experiments this information is disturbed bydetector effects like resolution and acceptance. Therefore to be able to compare Monte Carlopredictions with measurements we need to pass the output from a generator through asaccurate as possible simulation of the detector. The detector simulation is based on our bestknowledge of the geometry and of response of its different parts to incident particles. All theinformation about the response of the detector to the input from a generator we usually callthe detector level (of the Monte Carlo simulation). This detector level information is passedthrough the same reconstruction and analysis chains as the real data. Having informationon the generator and detector levels we are able to use Monte Carlo programs to correct thedata for detector acceptance and resolution effects and get insight into pure physics.

Usually even in a single analysis, several Monte Carlo generators are compared to thedata or used to correct the data for detector effects. Most relevant to the measurementspresented in this paper are the programs which are used to model two-photon interactionsfor double-tagged, single-tagged and anti-tagged events. The main features of the MonteCarlo programs used for analyses of LEP data are briefly described here. For further detailsthe reader is referred to the original publications. An overview can be found e.g. in [7,14,15].

PHOJET [35]: The general purpose Monte Carlo program, which is based on the DualParton Model [80] combined with the QCD improved parton model to give an almost com-plete description of hadron-hadron, photon-hadron and photon-photon interactions at highenergies. It contains both hard and soft processes. The hard processes are calculated in LOperturbative QCD, and soft processes are modeled based on γp, pp and pp data assumingRegge factorization. Originally only (quasi-)real photons were considered, but recently theprogram has been extended to match the deep inelastic electron photon scattering as well asthe scattering of two virtual photons. Both photons are allowed to fluctuate into a hadronicstate before they interact. The γ⋆γ or γ⋆γ⋆ cross-sections are obtained from the γγ cross-section by extrapolating in Q2 on the basis of the Generalized Vector Meson Dominancemodel using the description of Ref. [56]. In particular, it is worth pointing out here, that theprogram is not based on the cross section formula for deep inelastic electron-photon scatter-ing and always produces the same x and Q2 distributions, independent of the input structurefunction. Events are generated for both soft and hard partonic processes. A cut-off on thetransverse momentum of the scattered partons in the photon-photon centre-of-mass systemof 2.5 GeV is used to separate the two classes of events. For this reason the generation ofevents with W below 5 GeV is known to be incomplete. The hadronisation is based on theLund string model as implemented in JETSET [79].

HERWIG [59]: The general purpose Monte Carlo program, which has been upgradedto electron photon scattering during the LEP2 Workshop [14]. The most recent version

40

HERWIG5.9+kt (dyn) incorporates improvements in modeling of the intrinsic transversemomentum of the quarks within the photon. This version uses a modified transverse mo-mentum distribution, kt, for the quarks inside the photon, with the upper limit dynamically(dyn) adjusted according to the hardest scale in the event, which is of order Q2. In HER-WIG the photon flux is based on the EPA [18]. The hard interaction is simulated as eq→eqscattering, with the incoming quark generated according to a set of parton distribution func-tions. The incoming quark is subject to an initial state paron shower, which is designed insuch a way that the hardest emission is matched to the sum of the matrix elements for thehigher order resolved processes (g → qq, q → qg and γ → qq). The parton shower uses thetransverse momentum as evolution parameter and obeys angular ordering. This proceduredynamically separates events into point-like and hadron-like events, which can be differentfrom the choice made in the parton distribution functions. For hadron like events the pho-ton remnant gets a transverse momentum kt with respect to the direction of the incomingphoton, generated from a Gaussian distribution. The hadronisation is based on the clustermodel.

PYTHIA [57]: The general purpose Monte Carlo program, which can generate hard or softprocesses between leptons, hadrons and photons. In case of deep inelastic electron photonscattering, the quarks are generated according to parton distribution functions of the photon.The flux of the quasi-real photons has to be externally provided, and the correspondingelectron is only modeled in the collinear approximation. The program is based on the leadingorder matrix element for the process eq→eq. Higher order QCD processes are generated viaparton showers, using the parton virtuality as the evolution parameter. The separationinto point-like and hadron-like events is taken from the parton distribution functions. Forhadron-like events the photon remnant gets a transverse momentum kt generated from aGaussian distribution, and for point-like events the transverse momentum follows a powerlike behaviour, dk2

t /k2t , with kt,max = Q2 as the upper limit. The hadronisation is based on

the Lund string model.

TWOGAM [61]: The Monte Carlo program developed by the DELPHI Collaboration forthe simulation of two photon processes. TWOGAM generates three different subprocesses:QPM, QCD resolved photon processes and non-perturbative soft processes described by theVector Meson Dominance model [3]. The normalization of the QPM process is determinedby the quark masses. The normalizations of the VMD is fixed by the cross section forthe scattering of two real photons. The partons generated according to a chosen partondistribution function in the photon, undergo a hard 2 → 2 scattering process. No partonshowers are included. The hadronisation is based on the Lund string model. TWOGAMwas recently upgraded to take into account QED soft and hard radiation from initial andfinal state electrons.

Vermaseren [73]: The Monte Carlo program, which has become a standard calculationof the process e+e− → e+e− f f proceeding via γγ interactions. It uses the subset of theexact 2 → 4 matrix element in which the electron and positron do not annihilate. The fulldependence on the mass of final state fermion and an P 2 is kept.

41

BDK [66–68]: The program is an extension of the Vermaseren program and includes theQED radiative corrections to the process. It mainly serves to the estimation of the radiativecorrections to be applied to data corrected for acceptance with a different Monte Carloprogram which does not contain QED radiative corrections.

GALUGA [65]: The Monte Carlo program, which contains a full implementation of thecross section given in Eq.13. It is not used as an event generator, but due to its flexibility isoften used to calculate cross sections and photon fluxes in a user defined phase space.

42

7 The OPAL detector at LEP

The Large Electron Positron (LEP) collider was an e+e− storage ring located at CERN.It had a circumference of 27 km and was located 100 m underground. Four experiments:OPAL, ALEPH, DELPHI and L3, were situated at symmetrical collision points around thering. During the first phase of operation (LEP1), in the years 1989–1995, the centre-of-massenergy of the electron and positron beams was close to the mass of the Z0 gauge boson at91 GeV. The energy was increased for the LEP2 phase: first to 161 GeV to produce W+W−

pairs and then in steps up to 209 GeV by the end of operation in the year 2000.

A schematic view of the OPAL (Omni Purpose Apparatus for LEP) detector is shownin Fig.21. A detailed description of the OPAL detector can be found in Ref. [69]. Here onlya brief account is given of the main components relevant to the measurements in two photonphysics.The right-handed OPAL coordinate system is defined with the z axis pointing in the direc-tion of the e− beam and the x axis pointing towards the centre of the LEP ring. The polarangle θ, the azimuthal angle φ and the radius r are the usual spherical coordinates.

The central tracking system is located inside a solenoid magnet which provides a uniformaxial magnetic field of 0.435 T along the beam axis. The central tracking system consistsof a two-layer silicon micro-vertex detector [70], a high precision vertex drift chamber, alarge volume jet chamber with 159 layers of axial anode wires, and a set of z–chambers foraccurately measuring track coordinates along the beam direction. The transverse momentapT of tracks with respect to the z direction of the detector are measured with a precisionof σpT

/pT =√

0.022 + (0.0015 · pT)2 (pT in GeV) in the central region, θ > 753 mrad. Thejet chamber also provides energy loss, dE/dx, measurements which are used for particleidentification.

The central detector is surrounded in the barrel region (| cos θ| < 0.82) by a lead glasselectromagnetic calorimeter (ECAL) and a hadronic sampling calorimeter (HCAL). Outsidethe HCAL, the detector is surrounded by muon chambers. There are similar layers of de-tectors in the endcaps (0.81 < | cos θ| < 0.98). The barrel section consists of a cylindricalarray of 9440 lead-glas blocks with a depth of 24.6 radiation lengths. The endcap sections(EE) consist of 1132 lead-glas blocks with a depth of more than 22 radiation lengths. Thelectromagnetic energy resolution of the EE calorimeter is about 15%/

√E (E in GeV) at

polar angles above 350 mrad, but deteriorates closer to the edge of the detector.The small angle region from 47 to 140 mrad around the beam pipe on both sides of the

interaction point is covered by the forward detectors (FD) and the region from 25 to 59mrad by the silicon-tungsten luminometers (SW) [71]. During the LEP2 phase the lowerboundary of the SW acceptance was effectively 33 mrad due to the installation of a low-angleshield to protect the central detector from synchrotron radiation. The FD consists of cylin-drical lead-scintillator calorimeters with a depth of 24 radiation lengths divided azimuthallyinto 16 segments. The electromagnetic energy resolution is approximately 18%/

√E (E in

GeV). The SW detector consists of two cylindrical small angle calorimeters encircling thebeam pipe at approximately ±2.5 m from the interaction point. Each calorimeter is madeof a stack of 18 tungsten plates, interleaved with 19 layers of silicon sampling wafers and

43

θ ϕ

x

y

z

Hadron calorimeters and return yoke

Electromagnetic calorimeters Muon

detectors

Jet chamber

Vertex chamber

Microvertex detector

Z chambers

Solenoid and pressure vessel

Time of flight detector

Presampler

Silicon tungsten luminometer

Forward detector

Figure 21: Schematic view of the OPAL detector. Main components discussed in the textare indicated.

mounted as two interlocking C–shaped modules around the LEP beam pipe. The depth ofthe detector amounts to 22 radiation lengths. Each silicon layer consists of 16 wedge-shapedsilicon detectors. The sensitive area of the calorimeter fully covers radii between 81 and 142mm from the beam axis. The electromagnetic energy resolution is approximately 25%/

√E

(E in GeV) on both sides. At LEP2 energies it was found that the energy resolution isalmost constant with energy due to the energy leakage and dead material.Small angle detectors were primary used in the luminosity measurement, which was basedon the tagging of electrons and positrons scattered at small angles in the Bhabha process.

The other three experiments were constructed based on the similar general concepts,although they differ in technical and constructional solutions. For details see: ALEPH [108],DELPHI [109], L3 [110]. Also the geometrical (and in consequence kinematical) acceptancesof different subdetectors differ slightly between the experiments.

44

8 Studies of the photon and electron structure in DIS

8.1 QED structure functions of the photon

The measurements of the QED photon structure functions at e+e− colliders are possibleby studying the reaction e+e− → e+e− l+l−, where l = e, µ, τ , in deep inelastic electronphoton scattering regime, where one of the beam electrons scatters at sufficiently large polarangle to be observed in a detector, and the other remains undetected. Although from thetheoretical point of view, all lepton channels are equally well suited for this purpose, fromthe experimental side the most clean measurement can be performed with µ+µ− final state.The process has large cross section and is almost background free. For e+e− final state thecross section is even higher, but the number of different Feynman diagrams contributing tothis process makes the analysis in terms of the photon structure functions more difficult.The τ+τ− final state suffers from low statistics and can be only identified by detecting theproducts of τ decays. This makes the analysis quite difficult, because of large backgroundsfrom qq production or µ+µ− final state, depending on whether the hadronic or leptonic decaychannels of τ are used for its detection. The results presented in this section were obtainedfrom the measurement of the process e+e− → e+e− µ+µ−. In principle the measurement canserve as a test of QED to order O(α4). However, the experimental uncertainties are ratherlarge and the precision tests of QED are not the aim of the measurement. At e+e− collidersthis measurement rather serves as a test of experimental methods used in more difficultanalyses like the hadronic structure function of the photon.

The measurements of the QED structure functions of the photon at LEP base on datacollected during LEP1 phase and correspond in each experiment to an integrated luminosityclose to 100 pb−1. The photon structure function F γ

2,QED has been measured at LEP byOPAL [111], DELPHI [112] and L3 [113] experiments. The above measurements by OPALand DELPHI replaced the earlier measurements at lower statistics published in [117] andin [118]. There exist a preliminary measurement by ALEPH [119], which however is notgoing to be published and therefore is not considered here. Prior to LEP the measurementsof F γ

2,QED were performed by the CELLO [114], PLUTO [115] and TPC/2γ [116] experiments.The measurements of the structure function F γ

2,QED performed by the above experimentsare compared to the QED predictions in Fig. 22, and the values of F γ

2,QED together withavailable errors are listed in Tables 14, 15 and 16. If the average virtuality of the targetphoton 〈P 2〉 in the measurement is provided by the experiment, it is used in the QEDprediction. The value of 〈P 2〉 is usually obtained from a Monte Carlo program or fromthe best fit of the QED prediction to the data. All LEP experiments provide 〈P 2〉 of theirmeasurements and non of the experiments prior to LEP does it. In the later case P 2 = 0 wasassumed in the QED calculation. The value of the average virtuality of the probe photon 〈Q2〉or the range of Q2 used for F γ

2,QED measurement is given in Fig. 22 in parentheses close to thename of the experiment. In the case when 〈Q2〉 is provided, the measurement is comparedto the QED prediction for F γ

2,QED(x, 〈Q2〉, 〈P 2〉). In the case when the range of experimentalQ2 acceptance only is provided, the prediction for 〈F γ

2,QED(x, Q2, 〈P 2〉)〉 averaged over givenQ2 range is used instead. The data are generally in a good agreement with QED predictions.Only the measurement by TPC/2γ is systematically below the QED prediction. The reason

45

TPC/2γ(0.14-2.28)Fγ 2,QED /α

Fγ 2,

QE

D /α OPAL(2.2) L3(1.4-7.6) OPAL(4.2) PLUTO(5.5)

OPAL(8.4) CELLO(1.2-39) OPAL(12.4) DELPHI(12.5) OPAL(21)

PLUTO(40)

x

DELPHI(120) OPAL(130)

0

1

0

1

0

2

0 0.5 1

Figure 22: Compilation of all available measurements of F γ2,QED as a function of x. The

measured values of F γ2,QED/α are compared to the QED predictions. The statistical error

is given by the inner error bars, and the total errors as full error bars. The numbers inparentheses denote the average virtuality of the probe photon 〈Q2〉 or the range of Q2 inthe measurement. The QED predictions shown correspond to F γ

2,QED(x, 〈Q2〉, 〈P 2〉) or to〈F γ

2,QED(x, Q2, 〈P 2〉)〉 and use the average 〈P 2〉 value of the measurement if provided by theexperiment or P 2 = 0 as discussed in the text.

of this disagreement may be due to the averaging procedure used for the QED predictionas well as due to assumption that P 2 = 0, which might be of great importance especiallybecause of low virtuality of the probe photon in this measurement.

Another way of presentation of F γ2,QED is as a function of Q2, which is done in Fig. 23.

Data are grouped into bins of x of size of 0.1 (only the highest x bin is slightly narrowerdue to the upper kinematical limit) and central values 〈x〉 shown in the figure. To separatethe measurements in each bin of x an offset N indicated in the figure was added to eachmeasurement and QED prediction. The QED predictions shown in the figure represent anaverage F γ

2,QED in the x bin under study with P 2 = 0. All data points for which 〈P 2〉 isknown are corrected to P 2 = 0 by multiplying by the ratio of F γ

2,QED (or 〈F γ2,QED〉) calculated

at P 2 = 0 and at P 2 = 〈P 2〉. If for a particular measurement the range of Q2 is giventhen for each data point at given x a separate average value of Q2 has been calculatedbased on QED predictions for F γ

2,QED. This has the most significant influence on TPC/2γdata where the points at high values of x have been measured at effectively higher valuesof Q2 than the points at low values of x. The agreement between the measurements andthe QED predictions is very good in whole explored range of average photon virtualities0.43 < 〈Q2〉 < 130GeV2.

The process e+e− → e+e− µ+µ− has such a clear experimental signature that it ispossible to measure the triple differential cross section given in Eq. 43. The shape of the

46

Q2 [GeV2]

Fγ 2,

QE

D /α

+ N

0.05 1

TPC/2γ

0.05 1

0.15 20.15 2

0.25 30.25 3

0.35 40.35 4

0.45 5

0.55 6

0.65 7

0.75 8

0.85 9

0.935 10Fγ 2,QED(Q2, P2= 0) /α <x> N

OPAL

Fγ 2,QED(Q2, P2= 0) /α <x> N

L3

Fγ 2,QED(Q2, P2= 0) /α <x> NFγ 2,QED(Q2, P2= 0) /α <x> N

PLUTO

Fγ 2,QED(Q2, P2= 0) /α <x> NFγ 2,QED(Q2, P2= 0) /α <x> N

CELLO

Fγ 2,QED(Q2, P2= 0) /α <x> NFγ 2,QED(Q2, P2= 0) /α <x> N

DELPHI

Fγ 2,QED(Q2, P2= 0) /α <x> NFγ 2,QED(Q2, P2= 0) /α <x> NFγ 2,QED(Q2, P2= 0) /α <x> NFγ 2,QED(Q2, P2= 0) /α <x> NFγ 2,QED(Q2, P2= 0) /α <x> N

0

2

4

6

8

10

12

10-1

1 10 102

103

Figure 23: Compilation of all available measurements of F γ2,QED as a function of Q2. The

measured values of F γ2,QED/α are compared to the QED predictions. The statistical error is

given by the inner error bars, and the total errors as full error bars. The LEP data havebeen corrected for the effect of non-zero P 2. The curves represent the QED predictions forP 2 = 0. The average values of 〈x〉 and the offset N are indicated on the right side of thefigure.

distribution of the azimuthal angle χ (for the definition see Section 2):

dN

dχ= A

(

1 +F γ

A,QED

F γ2,QED

cos χ +1

2

F γB,QED

F γ2,QED

cos 2χ

)

(82)

can be used to determine the ratios of the structure functions for the real photon. Measuringin addition the structure function F γ

2,QED for the real photon, it is possible to obtain directlyF γ

A,QED and F γB,QED. The structure functions F γ

A,QED and F γB,QED themself or/and scaled by

the structure function F γ2,QED have been measured at LEP by OPAL [111], DELPHI [112]

and L3 [113] experiments. There exist no other measurements of these quantities. TheLEP results are listed in Tables 17 and 18. The measured ratios of the structure functionsF γ

A,QED/F γ2,QED and 1

2F γ

B,QED/F γ2,QED are compared in Fig. 24 between the experiments and

with theoretical predictions given by Eq. 44. The average virtuality of each measurement

47

x

FA

/Fγ 2

Q2= 5.4 GeV2OPALQ2=3.25 GeV2L3Q2=12.5 GeV2DELPHI

x

1/2

FB/F

γ 2

Q2= 5.4 GeV2OPALQ2=3.25 GeV2L3Q2=12.5 GeV2DELPHI

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.05

0

0.05

0.1

0.15

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

FA

/α

Q2= 5.4 GeV2OPALQ2=3.25 GeV2L3

x

FB/α Q2= 5.4 GeV2OPAL

Q2=3.25 GeV2L3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.05

0

0.05

0.1

0.15

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 24: The structure function ratios F γA,QED/F γ

2,QED and 12F γ

B,QED/F γ2,QED measured by

the OPAL, L3 and DELPHI experiments are compared to the predictions of QED. Thestatistical error is given by the inner error bars, and the total errors as full error bars. Thetick marks at the top of the figures indicate the bin boundaries of the OPAL measurement.The QED curves are drawn for each Q2 value separately.

Figure 25: The structure functions F γA,QED and F γ

B,QED measured by the OPAL and L3experiments are compared to the predictions of QED. The meaning of symbols as in Fig. 24.

is indicated in the figure, and the QED curves are drawn separately for each of them. Themeasurements agree well within the relatively large statistical errors between each otherand with QED predictions. For example, in case of the OPAL measurement, the QEDpredictions for the full range in x of F γ

A,QED/F γ2,QED = −0.027 and 1

2F γ

B,QED/F γ2,QED = 0.078

are well in agreement with the measured values F γA,QED/F γ

2,QED = −0.036 ± 0.027 ± 0.004

and 12F γ

B,QED/F γ2,QED = 0.061 ± 0.013 ± 0.004.

The strength of the χ dependence varies with the scattering angle cos θ⋆ of the muonsin the photon-photon centre-of-mass system. Reducing the acceptance of cos θ⋆ enhancesthe χ dependence, but to obtain a result for F γ

A,QED and F γB,QED which is valid for the

full range of cos θ⋆ the measurement has to be extrapolated using the predictions of QED.The measurements performed by L3 and DELPHI are obtained in the range | cos θ⋆| < 0.7and | cos θ⋆| < 0.94, respectively, and extrapolated to the full range in cos θ⋆. The OPALmeasurements are valid in the full angular range | cos θ⋆| < 1.The structure functions F γ

A,QED and F γB,QED have been extracted from the above ratios by

OPAL and L3 only. They are compared between each other and with the QED predictionsin Fig. 25. The agreement between both experiments and with QED predictions is good.Although the errors are large, it is clear from the measurements that both structure functionsF γ

A,QED and F γB,QED are different from zero and are not flat. As explained in Section 4, the

measurement of F γB,QED is a good estimate of the longitudinal structure function F γ

L,QED

which has been not directly measured sofar.

48

8.2 Hadronic structure function of the photon

The first measurement of the hadronic structure function of the photon F γ2 has been per-

formed by the PLUTO experiment [122]. Since then many other measurements have becomeavailable, with most precise results coming from the LEP experiments. The structure func-tion F γ

2 is extracted from the differential cross section for deep inelastic electron photonscattering process (Eq. 39). From the experimental point of view the measurements arebased on single tagged events with a hadronic final state. The most recent OPAL measure-ment of F γ

2 at low-x has been performed by the author of this paper. This section is dividedinto three parts. It starts with a more detailed description of the recent OPAL analyses atlow-x and at high Q2, in particular the event selection criteria, the background sources andthe description of data by Monte Carlo models is reviewed. In the second part all availablemeasurements of F γ

2 from LEP and from other experiments are presented and comparedbetween each other and with some recent parameterisations. In last the measurement of thecharm structure function F γ

2,c of the photon is discussed.

8.2.1 Measurement of F γ2 by the OPAL experiment at low-x

In this section we present the recent OPAL measurement of F γ2 [143]. The data sample

used in the analysis corresponds to an integrated luminosity of 592.7 pb−1 accumulated bythe OPAL experiment in 1998 (168.5 pb−1), 1999 (208.3 pb−1) and 2000 (215.9 pb−1) ate+e− centre-of-mass energies

√see = 189 − 209 GeV with a luminosity weighted average of√

see = 198 GeV.

Event selection and background estimation

As explained above, the measurement of the hadronic structure function F γ2 of the photon

is based on single tagged events with a hadronic final state. Such events were selected withthe following set of cuts (see Section 7 for more details on the OPAL detector):

1. An electron candidate should be observed in one of SW detectors, with energy E ′ >0.75Eb and polar angle in the range 34 < θ < 55 mrad. The angle θ is measured withrespect to the original beam direction. The calorimeter cluster with the highest energyis taken as the electron candidate. The high energy threshold for the electron candidateis needed to reject electron candidates from beam–gas interactions (off-momentumelectrons).

2. An anti-tag cut is applied for possible electron candidates in the hemisphere oppositeto the tag electron – there must be no cluster with energy Ea > 0.25Eb in SW detectoron the opposite side.

49

3. In order to remove events with scattered electrons in FD or in the central electromag-netic calorimeter, we require that there is no single cluster in these detectors with anenergy above 0.25Eb.

4. At least 3 tracks (Nch ≥ 3) must be found in the tracking system. A track is requiredto have a minimum transverse momentum of 120 MeV, and to fulfill standard qualitycuts as given in [72].

5. The visible invariant mass, Wvis, is required to be in the range of 2.5 GeV up to0.64Eb. The upper limit is dictated by the high background of annihilation eventsat large Wvis values. The quantity Wvis is reconstructed from tracks measured in thecentral tracking detectors and the position and energy of clusters measured in theelectromagnetic and hadronic calorimeters, as well as in the forward detectors FDand SW. A matching algorithm [72] is used to avoid double counting of the particlemomenta in the calorimeters and tracking chambers.

6. To reduce the remaining background due to beam–gas interactions, the z position ofthe primary vertex |〈z0〉| is required to be less than 4 cm from the nominal interactionpoint. Here 〈z0〉 is calculated as the error weighted average of the z coordinates of alltracks at the point of closest approach to the origin in the r−φ plane. We also requirethat the distance d0 of the primary vertex from the beam axis should be less than 0.5cm.

7. In order to ensure that the event is well contained in the detector and to reducebackground from beam–gas interactions, the total energy measured in the event mustbe less than 2.2Eb.

The numbers of events selected in each data taking period together with the numbers ofsignal events after subtracting the background contributions described below are given inTable 1. Only signal events, but split into two ranges of Q2 used for F γ

2 measurement, aregiven in Table 2. The average measured virtuality 〈Q2〉 is also given for each Q2 range andeach data taking period.Trigger efficiencies were evaluated from the data using sets of separate triggers and found tobe larger than 99% for the events within the selection cuts.

Several Monte Carlo generators have been compared to the data or have been used tocorrect the data for detector effects. For details on Monte Carlo programs mentioned belowsee Section 6. To simulate the signal events we use HERWIG 5.9+kt (dyn) [59], a generalpurpose Monte Carlo program that includes deep inelastic electron photon scattering. Theversion uses a modified transverse momentum distribution, kt, for the quarks inside thephoton, with the upper limit dynamically (dyn) adjusted according to the hardest scale inthe event, which is of order Q2. In order to have an additional model that contains differentassumptions for modeling the hard scattering and the hadronisation process the Monte Carloprogram PHOJET 1.05 was also used to simulate signal events. PHOJET simulates hardinteractions through perturbative QCD and soft interactions through Regge phenomenology,and the hadronisation is modeled by PYTHIA [57]. Several HERWIG and PHOJET sampleswere generated (with the integrated luminosity more than five times that of the data) using

50

year / luminosity 1998 / 168.5 pb−1 1999 / 208.3 pb−1 2000 / 215.9 pb−1

Data selected 7003 8414 8384Data signal 6447 ± 84 7745 ± 92 7657 ± 92HERWIG signal 6538 ± 16 7738 ± 19 8010 ± 20Backgroundsγ⋆γ → τ+τ− 349 ± 7.7 416 ± 9.3 446 ± 9.7γ⋆γ → e+e− 146 ± 5.0 174 ± 6.0 200 ± 6.7γ⋆γ⋆ → hadrons 27 ± 0.6 32 ± 0.7 32 ± 0.7Z0/γ

⋆ → hadrons 25 ± 0.9 36 ± 1.6 38 ± 2.04–fermion eeqq 6 ± 0.5 7 ± 0.6 8 ± 0.6

Table 1: The numbers of selected events and signal events (selected events corrected forbackground) in the data compared to the signal predictions from the HERWIG program.The expected numbers of background events for the dominant sources according to MonteCarlo are also listed. The errors given are only statistical.

Q2 range OPAL HERWIGyear

[GeV2 ] events 〈Q2〉 [GeV2 ] events 〈Q2〉 [GeV2 ] 〈Q2tru〉 [GeV2 ]

7 – 13 2576 ± 53 11.0 2653 ± 10 11.0 9.41998

13 – 27 3870 ± 65 17.7 3884 ± 12 17.7 18.37.5 – 14 3050 ± 57 11.9 2824 ± 12 12.0 10.1

199914 – 30 4695 ± 72 19.3 4914 ± 15 19.4 19.98 – 15 2876 ± 56 12.8 3008 ± 12 12.9 10.8

200015 – 33 4780 ± 73 20.7 5002 ± 16 20.8 21.6

Table 2: The numbers of signal events in the data compared to the signal predictions fromthe HERWIG program split into two bins in Q2. The average 〈Q2〉 values at the detectorlevel and true values of average 〈Q2

tru〉 predicted by HERWIG are listed.

the GRV LO [74] parameterisation of F γ2 , taken from the PDFLIB library [81], as the input

structure function. This version assumes massless charm quarks. Since PHOJET is notbased on the cross-section formula for deep inelastic electron photon scattering, the programalways produces the same x and Q2 distributions independent of the input structure function.Therefore the x distribution of PHOJET, for the use in unfolding procedure, was reweightedto match that from HERWIG using GRV LO. This is not an important limitation, becausethe main emphasis lies on the alternative hadronisation model. The result of the unfoldingprocedure is expected to be almost independent of the actual underlying x distribution ofthe Monte Carlo sample used. The number of expected signal events predicted by HERWIGare listed in Table 1. The same events, but split into two bins of Q2 together with averagemeasured and true 〈Q2〉 values are listed in Table 2. The measured values of 〈Q2〉 in dataand MC simulation agree well. The true values of 〈Q2〉 are slightly different because of thecuts applied - at the true level only cuts defining the bins in Q2 are applied. The averagevalues of Q2 for the F γ

2 measurement are obtained as the luminosity weighted average of thetrue values listed in Table 2.

51

The dominant background comes from the reaction e+e− → e+e− τ+τ− proceeding via themultiperipheral process shown in Fig. 3b, with e+e− → e+e− e+e− giving a smaller contribu-tion. These were simulated using the Vermaseren program [73]. The process Z0 → hadronsalso contributes significantly and was simulated using PYTHIA. The next largest back-ground comes from non-multiperipheral four-fermion events with eeqq final states (denotedby 4-fermion eeqq) and was simulated with GRC4f [62]. Because our aim is to measure thestructure function of the quasi-real photon, events stemming from the interaction of twovirtual photons with hadronic final states are also treated as background. These events weregenerated using PHOJET 1.10 with the virtualities of both photons to be above 4.5 GeV2.The contributions to the background from all other Standard Model processes was found tobe negligible. The expected numbers of events from the dominant background sources foreach data taking period are listed in Table 1.

None of the Monte Carlo programs used in this analysis contain radiative corrections tothe deep inelastic scattering process. These are dominated by initial state radiation from thedeeply inelastically scattered electron. The final state radiation is experimentally to a largeextent absorbed in the scattered electron energy measurement, due to the granularity ofthe calorimeters. The Compton scattering process contributes very little, and the radiativecorrections due to radiation of photons from the electron that produced the quasi-real targetphoton where shown to be small [82].In this analysis the radiative corrections have been evaluated using the TWOGAM 2.04Monte Carlo program [61], which includes initial state and final state radiation from bothelectron lines. The calculations are performed using mixed variables, which means that Q2 iscalculated from electron variables, while W 2 is calculated from hadronic variables. The valueof x is found from W 2 and Q2, exactly as for experimental analysis. Samples of TWOGAMwith radiative corrections switched on and off corresponding to the three average centre-of-mass energies of the data were generated separately. To each sample the experimentalrestrictions on the electron energy and polar angle, the minimum and maximum invariantmass and the anti-tag energy and angle were applied. The predicted ratio of the differentialcross section for each bin of the analysis is used to correct the data, i.e. the measured F γ

2

(as well as the corresponding differential cross section dσ/dx) are multiplied by the ratio ofthe non-radiative to the radiative cross sections. The radiative corrections with statisticalerrors, weighted with the luminosity of the data samples, are given in Table 3. In Fig. 26we show the value of the radiative corrections for the two ranges of Q2 as a function of xfor different subprocesses generated by TWOGAM as well as for the weighted average. Alsothe relative contribution of the individual subprocesses to the total cross section is shown inbottom plots. The radiative corrections are largely insensitive to the choice of F γ

2 used inthe calculation.

Several theoretical models exist for how F γ2 should behave as a function of P 2 [92, 100,

102]. They all predict a decrease of F γ2 (x, Q2, P 2 6= 0) with P 2 which is strongest at low

values of x. The size of the effect was studied using the GRSc [100] and SaS1D [102, 103]parameterisations. A stand alone analytical program has been used to calculate averagevalues of P 2 for each pair of (x, Q2) values considered in the measurement. The ratio ofF γ

2 (x, Q2, P 2 = 0) and F γ2 (x, Q2, 〈P 2〉) was used to obtain correction factors for the P 2 effect

in case of F γ2 and dσ/dx measurements. The errors on the correction factors are taken to

be the difference between the predictions from the GRSc and from the SaS 1D parameteri-

52

x(σ

born

-σra

d)/σ

rad

full

qpm

qcd

vmd

⟨Q2⟩=10.2GeV2

x

(σbo

rn-σ

rad)

/σra

d

⟨Q2⟩=20.0GeV2

x

rel.

frac

. [%

]

⟨Q2⟩=10.2GeV2

xre

l. fr

ac. [

%]

⟨Q2⟩=20.0GeV2

-7.5

-5

-2.5

0

2.5

5

7.5

10

12.5

10-3

10-2

10-1

1

-7.5

-5

-2.5

0

2.5

5

7.5

10

12.5

10-3

10-2

10-1

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

10-3

10-2

10-1

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

10-3

10-2

10-1

1

Figure 26: QED radiative corrections estimated with TWOGAM 2.04 Monte Carlo programin the phase space used for F γ

2 measurement in the latest OPAL analysis [143]. The relativedifference of non-radiative and radiative with respect to the radiative cross section is shownfor two ranges of Q2 as a function of x for different subprocesses generated by TWOGAMas well as for the weighted average. The relative contribution of the individual subprocessesto the total cross section is shown in bottom plots.

sations. Luminosity weighted average values of 〈P 2〉 and the correction factors are given inTable 3.

Properties of single-tagged γ⋆γ events

Scattered electrons in single tagged events are recorded on both sides of the OPAL detector.As the detector is fully symmetric there should be no statistically significant difference inany aspect between the events with electrons recorded on the left and on the right side. Thishas been checked to be the case.

A HERWIG Monte Carlo sample is used to correct the data for acceptance and resolutioneffects, and the PHOJET Monte Carlo is used to determine the systematic uncertainty onthis correction due to the different hadronisation model. A comparison is made of datadistributions with predictions from HERWIG and PHOJET. Variables calculated from thescattered electrons as well as variables calculated from the hadronic final state are studied.All Monte Carlo distributions shown here are normalized to the luminosity in data.

53

〈Q2〉 radiative bin-centre P 2 effect[GeV2 ]

x rangecor. [%] cor. [%] 〈P 2〉 [GeV2 ] cor. [%]

10.2 0.0009 - 0.0050 2.68 ± 1.37 3.61 ± 0.57 0.033 ± 0.005 9.81 ± 1.580.0050 - 0.0273 0.74 ± 0.57 -0.23 ± 0.62 0.042 ± 0.003 11.67 ± 0.830.0273 - 0.1496 -2.43 ± 0.35 -7.50 ± 0.82 0.075 ± 0.004 14.86 ± 0.180.1496 - 0.8187 -3.38 ± 0.25 10.55 ± 0.62 0.095 ± 0.002 10.75 ± 0.41

20.0 0.0015 - 0.0074 4.88 ± 1.43 2.25 ± 0.35 0.033 ± 0.001 9.37 ± 0.420.0074 - 0.0369 1.52 ± 0.51 -1.43 ± 0.63 0.052 ± 0.004 13.21 ± 0.380.0369 - 0.1827 0.61 ± 0.30 -7.70 ± 0.87 0.092 ± 0.004 15.27 ± 0.140.1827 - 0.9048 0.21 ± 0.20 6.06 ± 0.90 0.116 ± 0.002 9.24 ± 0.30

Table 3: The values of the corrections which have been applied to data, averaged overdifferent data taking periods, are listed. For a given bin in Q2 and x the correction isthe difference of the corrected and non corrected cross sections as a percentage of the noncorrected cross section. Average values of P 2 in each bin are also given. For the details ofhow the correction have been estimated and the meaning of the errors see text.

In Fig. 27 the energy scaled by the beam energy, azimuthal and polar angles of thetagged electrons are shown. All distributions are reasonably well described by the sum ofthe signal as predicted by HERWIG or PHOJET and the estimated background from otherphysics processes. In Fig. 27 also distributions are shown which characterize the hadronicfinal state: the number of tracks, Nch, the hadronic energy, Ehad, and the visible hadronicinvariant mass, Wvis. All the distributions are again reasonably well described by the sumof the signal as predicted by HERWIG and the estimated background from other physicsprocesses. The description by the PHOJET Monte Carlo is somewhat less good.

In Fig. 28 we show the distributions of the missing longitudinal, ∆pz, and transverse,∆pT momentum. The sum runs over all visible tracks and calorimeter deposits and includethe scattered electron and the electron lost in the beam pipe (for which the momentumof the respective beam was assumed). Fig. 28 also shows the missing longitudinal andtransverse momenta for the hadronic final state alone. Finally Fig. 28 shows the distributionof the transverse component of the energy out of the tagging plane. In all HERWIG plusbackground describes these distributions reasonably well, while the description is less goodfor PHOJET plus background.

In Fig. 29 we show the distributions of the following kinematical variables: the virtualityof the probe photon, Q2, and the dimensionless scaling variables defined in Section 2, ye, zand x for tagged events. All these distributions are well described by the sum of the signalas predicted by HERWIG or PHOJET and the background processes.

Finally Fig. 30 shows the distributions of the energy and transverse energy flows insingle tagged events averaged over all selected events. The polar angle in the definition ofpseudorapidity, η = − ln tan θ/2, is always measured with respect to the direction of thatoriginal beam electron which produced the tag electron in the detector. The azimuthalangle of the particles is always calculated with respect to the electron tagging plane. TheMonte Carlo predictions are obtained as the weighted sums of HERWIG or PHOJET andthe background processes, which are close to the predictions of HERWIG or PHOJET alone,within the cuts used in this analysis. In all HERWIG gives a good description of the data

54

E/Ebev

ents

OPALHERWIG+bgr.PHOJET+bgr.γ✭ γ→τ+τ-

γ✭ γ→e+e-

other bgrs.

a)

Nch

even

ts

b)

φ [rad]

even

ts

c)

Ehad [GeV]

even

ts

d)

θ [rad]

even

ts

e)

W [GeV]

even

ts

f)

0

2000

4000

6000

8000

0.4 0.6 0.8 10

2000

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0 5 10 15 20

0

1000

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0 2 4 60

2500

5000

7500

10000

0 20 40

0

2000

4000

6000

8000

0.03 0.04 0.051

10

10 2

10 3

10 4

0 20 40 60

Figure 27: Distributions of a) the electron energy normalized to the energy of the beamelectrons, b) the number of charged tracks, c) the electron azimuthal angle, d) the visiblehadronic energy, e) the electron polar angle, f) the visible hadronic invariant mass, for se-lected single tagged events. The histograms are the predictions from HERWIG and PHOJETwith background added and for the background itself. The errors are statistical only.

while PHOJET gives only a fair description. In particular, PHOJET predicts to much energyin the central region of pseudorapidity. Also in the azimuthal angle PHOJET underestimatethe hadronic energy spread in the direction opposite that of the tagged electron.

Results

The aim of the analysis is the extraction of the photon structure function F γ2 from the dif-

ferential cross section given in Eq. 39. To measure the differential cross section dσee/dx dQ2

we perform a two dimensional unfolding using the GURU program [90]. As the variables forthe unfolding we have chosen x and Q2. For backward compatibility we use the same bins inx as in the previous OPAL paper [75]. In the variable Q2 we use four bins between 4.5 GeV2

and 50 GeV2 with the two inner bins equal to the ranges used for F γ2 measurement and given

55

∆pz/Ebev

ents

a)

∆pT/Eb

even

ts

b) OPALHERWIG+bgr.PHOJET+bgr.γ✭ γ→τ+τ-

γ✭ γ→e+e-

other bgrs.

∆pzh/Eb

even

tsc)

∆pTh/Eb

even

ts

d)

Eout/Eb

even

ts

e)

pout/Eb

even

ts

f)

0

2000

4000

-0.5 0 0.50

5000

10000

0 0.02 0.04 0.06 0.08 0.1

0

2000

4000

6000

-0.5 0 0.50

2000

4000

6000

8000

0 0.02 0.04 0.06 0.08 0.1

0

2000

4000

6000

8000

0 0.05 0.1 0.150

2500

5000

7500

10000

-0.1 -0.05 0 0.05 0.1

Figure 28: Distributions of missing transverse (∆pT ) and longitudinal (∆pz) momentum forall particles (a,b), for hadrons only (c,d), and the scalar and vector sum of the energy com-ponent out of the tag plane (e,f), with HERWIG and PHOJET predictions and backgroundestimates. All quantities shown are normalized to the energy of the beam electron. Theerrors are statistical only.

in Table 2. We obtain the photon structure function F γ2 by dividing the differential cross

sections dσee/dx dQ2 by the flux calculated according to the Eq. 38 and the appropriatekinematical term. The unfolded F γ

2 and the cross section dσ/dx are further corrected forthe QED radiative effects and for the P 2 effect. Bin-centre correction has been in additionapplied to F γ

2 . Bin-centre correction is calculated as the average of GRV LO and SaS1Dpredictions for the correction from the average F γ

2 over the bin in x and the range of Q2 tothe value of F γ

2 at the nominal (x,〈Q2〉) position. The luminosity weighted average values ofbin-centre corrections and the nominal values of x and 〈Q2〉 are given in Table 3. The errorson the bin-centre correction are obtained as a half of the difference between the predictionsof GRV LO and SaS1D parameterisations.

The systematic error of the measurement has been evaluated taking into account severalcontributions. All changes of selection cuts, except the change of the cut on Nch, are appliedto the data. The list of systematic checks include:

56

Q2 [GeV2]

even

ts

a)

ye

even

ts

OPALHERWIG+bgr.PHOJET+bgr.γ✭ γ→τ+τ-

γ✭ γ→e+e-

other bgrs.

b)

log(z)

even

ts

c)

log(x)

even

ts

d)

0

2000

4000

6000

8000

10000

0 10 20 30 400

2000

4000

6000

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10000

12000

14000

0 0.5 1

0

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2000

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4000

5000

6000

7000

8000

9000

-4 -3 -2 -1 01

10

10 2

10 3

10 4

-3 -2 -1 0

η

dET/d

η [G

eV]

a)

φ [rad]

dET/d

φ [G

eV/r

ad]

b) OPALHERWIG+bgr.PHOJET+bgr.

η

dE/d

η [G

eV]

c)

φ [rad]

dE/d

φ [G

eV/r

ad]

d)

0

0.5

1

1.5

2

2.5

3

3.5

4

-4 -2 0 2 40

1

2

3

4

5

6

-4 -2 0 2 4

00.5

11.5

22.5

33.5

44.5

5

-4 -2 0 2 40

1

2

3

4

5

6

7

8

-4 -2 0 2 4

Figure 29: Distribution of kinematic quantities: a) Q2, b) ye, c)log(z) and d) log(x). Thehistograms are the predictions from HERWIG and PHOJET with background added andfor the background itself. The errors are statistical only.

Figure 30: Distribution of transverse energy (a,b) and energy (c,d) flows in single taggedevents versus pseudorapidity η and azimuthal angle φ. The polar angle in the definitionof pseudorapidity is always calculated with respect to the direction of that beam electronwhich produced the tagged electron. The azimuthal angle is calculated with respect to theelectron tagging plane. The histograms are the weighted sums of predictions of HERWIGor PHOJET and background processes. The errors are statistical only.

1. The dependence on the Monte Carlo model used in the unfolding has been estimatedby repeating the unfolding using the PHOJET sample. Half of the difference betweenthe results of unfolding based on HERWIG and on PHOJET has been taken as thesystematic error, both for the positive and negative error.

2. The error due to a possible shift of the energy scale of the SW detectors was taken intoaccount by scaling the electron energy by ±1%, in accord with the uncertainty in thescale observed in single-tagged events, which is conservative because the differencesobserved for Bhabha events were much smaller.

3. The uncertainty in the description of the energy scale of the central electromagneticcalorimeter ECAL was taken into account by varying the energy scale by ±3% [75].

4. The uncertainty of the position of SW detectors in the plane perpendicular to the z-axis has been estimated changing the lower cut on θ by ±0.2 mrad. This was estimatedusing back-to-back topology of Bhabha events.

5. The uncertainty in the description of the interaction point was estimated by changingthe cut on |〈z0〉| by ±1.0 cm and the cut on the distance of the primary vertex fromthe beam axis was changed by ±0.1 cm.

57

6. The uncertainty in the measurement of hadronic energy was estimated by changingthe lower cut on W ±0.2 GeV and the upper cut by ±3 GeV.

7. The cut on the number of charged tracks has been changed from Nch ≥ 3 to Nch ≥ 4.This change has been applied simultaneously to data and Monte Carlo.

8. The number of bins used for the measured variable can be different from the numberof bins used for the true variable (though it should be at least as large). The standardresult has 6 bins in the measured x variable. This was increased to 8 to estimate thesystematic effects of unfolding.

The final results for F γ2 and the cross section dσ/dx are the average of the results obtained

from unfolding using HERWIG and PHOJET. The total systematic error was obtained byadding in quadrature errors coming from the above contributions. The sum has been per-formed separately for positive and negative contributions. The errors from the QED radiativecorrection, correction for the P 2 effect as well as of the bin-centre correction have been addedin quadrature to the total systematic error. In Table 5 we present the individual contribu-tions to the total systematic error. The numbers are given as percentage of the nominalvalue, and are averaged over the different data taking periods. The main contribution tothe systematic error comes from the modeling of the hadronic final state as predicted byHERWIG and PHOJET. In the lower Q2 range the results are also quite sensitive to theprecise determination of the position of SW detectors, and in the higher Q2 range the energyscale in SW detectors plays a more significant role.

In Fig. 31 we present the measurements of the photon structure function in two bins ofQ2, and compare it with the results of the previous OPAL publication [75]. For this figurethe P 2 correction, which is about 10%, was not applied, such that these measurements canbe compared. The results agree well with the previous data. In Figs. 32 the measurement ispresented including the correction for the target photon virtuality. The results are comparedto predictions from structure functions based on different parton distribution functions. Theresults are compatible with a rise of F γ

2 with decreasing x. In Table 4 we give the measureddifferential cross sections for the process e+e− → e+e−γ⋆γ → e+e−X and the photon struc-ture function F γ

2 as a function of x in two bins of Q2. The discussion of the results presentedhere, together with the results on F γ

2 from other experiments will be given in Section 8.2.3.

58

x

F2γ (x

)/α

⟨Q2⟩=10.2 GeV2

OPAL98 publ.

OPALOPALOPAL

x

F2γ (x

)/α

⟨Q2⟩=20 GeV2

LAC1CJKLGRSc LOGRV LO

WHIT1SaS1DQPM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10-3

10-2

10-1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10-3

10-2

10-1

x

F2γ (x

)/α

⟨Q2⟩=10.2 GeV2

OPALOPALOPAL

x

F2γ (x

)/α

⟨Q2⟩=20 GeV2

LAC1CJKLGRSc LOGRV LO

WHIT1SaS1DQPM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10-3

10-2

10-1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10-3

10-2

10-1

Figure 31: Photon structure function F γ2 measured by the OPAL experiment in this analysis

(full circle), not corrected for the target photon virtuality P 2, compared to the previouslypublished result (open circle), shown in two bins of Q2. For each point, the inner error barsshow the statistical error and the full error bars show the total error. Predictions of differentparameterisations of F γ

2 are shown as curves.

Figure 32: Photon structure function F γ2 measured by the OPAL experiment, corrected for

the virtuality of the target photon P 2, shown in two bins of Q2. For each point, the innererror bars show the statistical error and the full error bars show the total error. Predictionsof different parameterisations are shown as curves.

〈Q2〉 x

[GeV2 ] range bin-centredσ/dx [pb] F γ

2 /α

10.2 0.0009 - 0.0050 0.0021 2098 ± 62 +171−192 0.519 ± 0.015 +0.040

−0.045

0.0050 - 0.0273 0.0117 682 ± 14 +49−57 0.340 ± 0.007 +0.021

−0.026

0.0273 - 0.1496 0.0639 215 ± 4 +14−15 0.315 ± 0.005 +0.020

−0.021

0.1496 - 0.8187 0.3143 74 ± 1 +8−7 0.435 ± 0.008 +0.040

−0.040

20.0 0.0015 - 0.0074 0.0033 768 ± 24 +80−94 0.569 ± 0.017 +0.063

−0.073

0.0074 - 0.0369 0.0166 287 ± 6 +20−24 0.374 ± 0.008 +0.026

−0.031

0.0369 - 0.1827 0.0821 110 ± 2 +4−6 0.379 ± 0.006 +0.016

−0.019

0.1827 - 0.9048 0.3483 43 ± 1 +4−4 0.530 ± 0.008 +0.043

−0.042

Table 4: Results on the measurements of dσ/dx and F γ2 /α for two different values of photon

virtuality 〈Q2〉. In each case the first error is statistical and the second systematic. Thevalues of the cross section are the average values in the bin, and the values of F γ

2 are correctedto the bin centre.

59

〈Q2〉 x bin total error

[GeV2 ] centreF γ

2 /αstat. sys.

θmin− θmin+ ESW+ ESW− Ehad+ Ehad− Wmin− Wmin+ Wmax+ Wmax

10.2 0.0021 0.519 2.9 +7.7−8.7 4.72 −4.36 −0.65 −0.56 1.22 −1.54 −2.20 0.32 0.14 −0.29

0.0117 0.340 2.1 +6.3−7.6 4.99 −4.74 −1.10 0.53 0.89 −0.97 −1.79 −0.91 0.08 −0.16

0.0639 0.315 1.6 +6.3−6.7 5.25 −5.14 −1.71 2.01 0.43 −0.23 1.47 −2.60 0.02 −0.04

0.3143 0.435 1.8 +9.2−9.2 5.67 −5.55 −2.28 3.41 0.03 0.41 7.53 −4.28 −0.02 0.03

20.0 0.0033 0.569 3.0 +11.1−12.8 −0.24 0.23 2.82 −4.25 2.20 −2.29 −3.67 0.64 0.15 −0.23

0.0166 0.374 2.1 +7.0−8.3 −0.27 0.26 2.78 −3.78 1.49 −1.59 −1.88 −0.35 0.08 −0.13

0.0821 0.379 1.6 +4.2−5.0 −0.30 0.31 2.81 −3.17 0.47 −0.56 1.68 −1.90 0.02 −0.03

0.3483 0.530 1.5 +8.1−7.9 −0.34 0.36 2.85 −2.86 −0.56 0.49 5.17 −3.60 −0.02 0.03

〈Q2〉 x bin total error MC unfol. bin P 2 rad.

[GeV2 ] centreF γ

2 /αstat. sys. model bins centre corr. corr. Nch > 3 |z0|− |z0|+ |d0|− |d0|

10.2 0.0021 0.519 2.9 +7.7−8.7 ±5.20 −0.98 ±0.57 ±1.58 ±1.37 −1.23 −1.90 1.25 −0.23 0.17

0.0117 0.340 2.1 +6.2−7.6 ±2.65 −0.74 ±0.62 ±0.83 ±0.57 −0.66 −2.41 1.56 −0.31 0.25

0.0639 0.315 1.6 +6.3−6.7 ±0.63 −0.28 ±0.82 ±0.18 ±0.35 0.19 −3.05 1.97 −0.42 0.34

0.3143 0.435 1.8 +9.2−9.2 ±4.37 −0.08 ±0.62 ±0.41 ±0.25 0.81 −3.62 2.39 −0.54 0.47

20.0 0.0033 0.569 3.0 +11.1−12.8 ±10.02 −0.68 ±0.35 ±0.42 ±1.43 −0.16 −2.18 1.34 −0.38 0.14

0.0166 0.374 2.1 +7.0−8.3 ±5.61 −0.50 ±0.63 ±0.38 ±0.51 −0.38 −2.50 1.53 −0.45 0.20

0.0821 0.379 1.6 +4.2−5.0 ±1.06 −0.13 ±0.87 ±0.14 ±0.30 −0.70 −2.95 1.82 −0.57 0.29

0.3483 0.530 1.5 +8.1−7.9 ±5.47 0.33 ±0.90 ±0.30 ±0.20 −0.96 −3.41 2.15 −0.69 0.36

Tab

le5:

System

aticerrors

forF

γ2/α

asa

fun

ctionof

xin

bin

sof

Q2.

Th

estatistical

erroran

dth

econ

tribu

tions

toth

esy

stematic

errorfrom

eachof

the

sources

arelisted

asa

percen

tagesof

Fγ2/α

.

60

8.2.2 Measurement of F γ2 by the OPAL experiment at high Q2

The OPAL measurements of the hadronic structure function of the photon F γ2 have been

recently extended to an average Q2 of 〈Q2〉 = 780 GeV2 using data collected at e+e− centre-of-mass energies from 183 to 209 GeV, with a luminosity weighted average of

√see = 197.1

GeV, taken in the years 1997-2000. The measurement is based on single tagged events withthe scattered beam electron recorded at very high polar angles in the electromagnetic endcapcalorimeters (EE). Here only the main results of the analysis will be given, and for moredetails the reader is referred to Ref. [140].

The single tagged events are selected by applying cuts on the scattered electrons and onthe hadronic final state. The scattered electron is required to have energy E ′ > 0.7Eb andpolar angle in the range 230 < θ < 500 mrad. To remove electron candidates originatingfrom energetic electromagnetic clusters stemming, e.g., from hadronic final states in thereaction Z0/γ⋆ → hadrons, an isolation cut is applied which requires that less than 3 GeVis deposited in cone of 500 mrad half-angle around the electron candidate. To ensure thatthe virtuality of the quasi-real photon is small, the highest energy electromagnetic cluster inthe hemisphere opposite to the one containing the scattered electron must have an energyEa < 0.25Eb. The number of tracks must be at least four, of which at least two tracksmust not be identified as electrons. The visible invariant mass is required to be in the range2.5 < Wvis < 60 GeV. The quality of the description of the data by Monte Carlo models inthis kinematical region is shown in Fig. 33. The data distributions of the energy and polarangle of the scattered electron are well described by both HERWIG and Vermaseren MonteCarlo models. For the variables related to the hadronic final state, the number of tracks Ntrk

and the visible hadronic invariant mass Wvis, there are apparent differences in shape.The data have been unfolded using three bins in x spanning the range 0.15 − 0.98 and

for 400 < Q2 < 2350 GeV2. The central values have been obtained using HERWIG andthe RUN program [91] for the unfolding. Each data point is corrected for radiative effectsusing the RADEG program [89]. Bin-centre corrections are also applied as given by theaverage of the GRSc [100], SaS 1D [102], and WHIT1 [94] predictions for the correctionfrom the average F γ

2 over the bin to the value of F γ2 at the nominal x position at the bin

centre. The measured F γ2 /α, shown in Fig. 34 together with several theoretical predictions,

exhibits a flat behaviour. It can be seen that in this high Q2 regime the differences betweenthese predictions are moderate, particularly in the central x-region. All these predictions arecompatible with the data to within about 20%, with the WHIT1 parameterisation, whichpredicts the flattest behaviour, being closest to the data. The QPM curve, which modelsonly the point-like component of F γ

2 , is calculated for four active flavours with masses of0.325 GeV for light quarks and 1.5 GeV for charm quarks. This prediction shows a muchsteeper behaviour in x and is disfavoured by the data.

The same data have been used in [140] to measure F γ2 at lower Q2 but still only for

x > 0.1. They are used together with the high Q2 measurement presented above to studythe evolution of F γ

2 with Q2. The results are shown in Fig. 35. The data in Fig. 35a exhibitpositive scaling violations in F γ

2 for the x ranges 0.10 − 0.25 and 0.25 − 0.60. For the range0.60 − 0.85, the data are compatible with the predicted scaling violations. To quantify theslope for medium values of x, where data are available at all values of Q2, the data were

61

0

10

20

30

40

50

60

70

80

90

0.6 0.8 1 1.20

20

40

60

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200 300 400 500

0

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5 10 150

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20 40 60

Etag/EbE

vent

s

(a)

θtag [mrad]

Eve

nts

(b)OPAL EEHERWIG 5.9+kt(dyn)Vermaseren

Z0/γ∗ → τ+τ−

γ∗ γ → τ+τ−

4-fermion eeqqZ0/γ∗ → hadrons

Ntrk

Eve

nts

(c)

Wvis [GeV]E

vent

s

(d)

Figure 33: Comparison of OPAL data distributions of the single tagged events recorded in thecentral electromagnetic calorimeter (EE) with Monte Carlo predictions. The open histogramsare the sum of the signal prediction and the contributions of the major background sources,shown for both HERWIG (solid line) and Vermaseren (dash-dotted line) models. The MonteCarlo predictions are normalised to the data luminosity. The distributions are: a) Etag/Eb,the energy of the scattered electron as a fraction of the energy of the beam electrons; b) θtag,the polar angle of the scattered electron; c) Ntrk, number of tracks; d) Wvis, the measuredinvariant mass of the hadronic final state. The errors given are only statistical.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

Fγ 2(

x)/α

OPAL, <Q2>=780 GeV2

GRScSAS1D

WHIT1QPM

Figure 34: The measured F γ2 /α as a function of x for 〈Q2〉 = 780 GeV2. The data are

compared to the predictions from GRSc (full line), SaS1D (dotted line), WHIT1 (dashedline) and QPM (dot-dashed line) parameterisations of F γ

2 /α. The inner error bars representthe statistical errors and the outer error bars represent the statistical and systematic errorsadded in quadrature.

62

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

1 10 102

103

Q2 [GeV2]

Fγ 2(

Q2 )/

α +

offs

et (a)

offset = N*0.45N x range

2 0.6-0.85

1 0.25-0.6

0 0.1-0.25

OPALGRScSAS1D

WHIT1QPM

0

0.2

0.4

0.6

0.8

1

1 10 102

103

Q2 [GeV2]

Fγ 2(

Q2 )/

α (b)OPAL √see = 183-209 GeVOPAL √see = 91 GeVOPAL √see = 161/172 GeV

0.1 < x < 0.6GRV HOSAS1DWHIT1QPMFit

Figure 35: The evolution of F γ2 /α as a function of Q2 for several bins of x, (a) 0.10 − 0.25,

0.25 − 0.60 and 0.60 − 0.85 and (b) for the central region 0.10 − 0.60. The inner errorbars represent the statistical errors and the outer error bars represent the statistical andsystematic errors added in quadrature. In (a) the data are compared to the predictions fromGRSc (full line), SaS1D (dotted line), WHIT1 (dashed line) and QPM (dot-dashed line)parameterisations of F γ

2 /α. In (b) GRSc has been replaced by the higher order predictionfrom GRV and, in addition, the result of the fit discussed in the text is shown.

fitted in the region 0.1 < x < 0.6, with a linear function of the form a + b ln Q2, where Q2 isin GeV2, resulting in:

F γ2 (Q2)/α = (0.08 ± 0.02+0.05

−0.03) + (0.13 ± 0.01+0.01−0.01) ln (Q2/GeV2)

with a correlation between the two parameters of −0.98 and a χ2/ndf of 10/3.

The data, together with the fit result, are shown in Fig. 35b. The result of the fit is inagreement and has significantly reduced errors on the parameters a and b compared to the

63

previous OPAL measurement of the Q2 evolution of F γ2 , published in [138]:

F γ2 (Q2)/α = (0.16 ± 0.05+0.17

−0.16) + (0.10 ± 0.02+0.05−0.02) ln (Q2/GeV2)

Both for the measurement of F γ2 at 〈Q2〉 = 780 GeV2 and for the investigation of the Q2

evolution of F γ2 , the quark parton model (QPM) is not in agreement with the data. It shows

a much steeper rise than the data as a function of x for 〈Q2〉 = 780 GeV2 and also a differentbehaviour in the Q2 evolution. In contrast the leading order GRSc, SaS 1D and WHIT1parameterisations and the higher order GRV parameterisation of F γ

2 are much closer to thedata. This means that the corresponding parton distribution functions of the photon areadequate to within about 20% at large values of x and at 〈Q2〉 scales of about 780 GeV2.

8.2.3 World results on the measurements of F γ2

In this section all presently available results on F γ2 from LEP experiments as well as from

experiments at other e+e− colliders are collected. The structure function F γ2 is presented

as a function of both x and Q2. The range of 〈Q2〉 covered by the various experiments is0.24 < 〈Q2〉 < 780 GeV2, and the measurements reach as low x as approximately 10−3. Ofspecial interest is the low-x behaviour of F γ

2 in comparison to the proton structure functionF p

2 . The large span of of the measurements as a function of Q2 allows the investigationof the evolution of F γ

2 with Q2. The results presented here are only the published resultsfrom the non LEP experiments and the published or preliminary (which are likely to bepublished soon) results from the LEP experiments. The following measurements are takeninto account: AMY [123, 124], JADE [125], PLUTO [126, 127, 145], TASSO [128], TPC/2γ[129], TOPAZ [130], ALEPH [131], DELPHI [133], L3 [135, 136] and OPAL [137–140, 143].Numerical values of all measurements of F γ

2 together with available errors are collected inTables 20-24.

A summary of all measurements of F γ2 as a function of x is shown in Fig. 36. The value

of 〈Q2〉 for each measurement is given in parentheses close to the name of the experiment.The data are compared to the predictions of the GRV HO and SaS1D parameterisationsof F γ

2 , drawn for the average Q2 of the measurements shown in a particular plot. Theparameterisations reflect the general behaviour of the data, however, there are regions wherethe agreement is not good. For example, the TPC/2γ data show quite unexpected shape inthe low x region. This strange behaviour might be due to underestimated systematic errorsin the TPC/2γ measurements. The measurements from the TOPAZ experiment rise fastertowards low values of x than newer and more precise data from LEP. Also the measurementsby TASSO and JADE, in spite of large errors, seem to be systematically shifted with respectto the theoretical predictions and newer LEP measurements. To get a quantitative estimationof the agreement between predictions of F γ

2 and the data we use simple χ2 method:

χ2 =ndf∑

i=1

(

F γ2,i − F γ

2,par

σi

)2

(83)

where the sum runs over all data points. The term F γ2,i denotes the i-th measured value of

F γ2 and σi is its total error. The theoretical prediction F γ

2,par of a given parameterisation

64

is approximated by F γ2 (〈x〉, 〈Q2〉, 0) or 〈F γ

2 (x, 〈Q2〉, 0)〉 depending on whether the measuredvalue of F γ

2 is given at a point 〈x〉 or averaged over a bin in x. If systematic errors quotedby an experiment are asymmetric, than the value chosen depends on whether the predictionlies below or above the measured value. The above procedure does not take into accountthe correlations of the errors between the data points and the experiments. However, asmentioned in [7], because there are common sources of errors, it is most likely that byusing this method the experimental errors are overestimated, and when using a more precisemethod, the predictions which are not compatible with the data would give even worseapproximations of the data.The data are compared to the predictions of GRV LO, GRV HO, GRSc, SaS1D, SaS 1M andthe most recent CJKL parameterisations of F γ

2 . The results of this comparison are listed inTable 6. The number of data points used in the comparison depends on the lowest valueof Q2 for which a parameterisation of F γ

2 is valid. All measurements prior to LEP, but theTPC/2γ data points, are well described by the selected parameterisations. The TPC/2γmeasurements which show an unexpected behaviour as a function of x are disfavoured byall parameterisations. The LEP measurements by OPAL, ALEPH and L3 are in goodagreement with the predictions. In all cases the best value of χ2/ndf give the ALEPHmeasurements. The measurements by the DELPHI experiment, which tend to rise fastertowards lower values of x, than other measurements, give the worst agreement with thepredictions. The data from most of experiments separately as well as combined data pointsfrom all experiments are in best agreement with the CJKL parameterisation, which is themost recent parameterisation available. The data at higher values of Q2 are better describedby the existing parameterisations than the data at lower values of Q2.

The same data as in Fig. 36 are shown as a function of Q2 in Fig. 37. The dataare compared to the predictions of the GRV HO and SaS1D parameterisations of F γ

2 . Themeasurements are separated into bins of x, with bin boundaries of 0.001, 0.01, 0.1, 0.2,0.3, 0.4, 0.6, 0.8, 0.99, and central values 〈x〉 indicated in the figure. To separate themeasurements in different bins of x, an offset value N has been added to each measurementand theoretical prediction. The theoretical predictions are obtained as the average F γ

2 /αin the bin. The general behaviour of the data is followed by the predictions. As expected,the photon structure function F γ

2 exhibits positive scaling violations for all values of x, withthe slope slightly increasing with increasing x. The kinks in the theoretical predictions arecaused by the charm threshold. The charm quark contribution to F γ

2 gets larger at highervalues of Q2.To study the Q2 evolution of F γ

2 usually data are fitted in the medium range of x, with alinear function of the form a + b ln Q2, where Q2 is in GeV2. The recent fit of that form tothe OPAL data in the region 0.1 < x < 0.6 and 7.5 < Q2 < 780 GeV2, was discussed in theprevious section, and reads:

F γ2 (Q2)/α = (0.08 ± 0.02+0.05

−0.03) + (0.13 ± 0.01+0.01−0.01) ln (Q2/GeV2)

A similar study has been performed by the L3 experiment in [136]. The data were fitted intwo regions of x, 0.01 < x < 0.1 and 0.1 < x < 0.2, using the Q2 range of 1.2 − 30 GeV2.The results of the fit are:

F γ2 (Q2)/α = (0.13 ± 0.01 ± 0.02) + (0.080 ± 0.009 ± 0.009) ln (Q2/GeV2) in 0.01 < x < 0.1

F γ2 (Q2)/α = (0.04 ± 0.08 ± 0.08) + (0.13 ± 0.03 ± 0.03) ln (Q2/GeV2) in 0.1 < x < 0.2

65

χ2/ndfExp. name ndf

GRV LO GRV HO GRSc SaS1D SaS 1M CJKL

AMY 8 0.75 1.15 0.73 0.98 0.91 0.65

JADE 8 1.01 1.16 1.01 1.09 1.00 0.79

PLUTO 13 0.50 0.46 0.60 0.60 0.51 1.22

TASSO 5 0.97 0.76 1.16 1.03 0.85 1.29

TOPAZ 8 2.02 2.41 1.87 2.53 2.03 1.77

TPC/2γ 19/15/11 4.67 4.27 7.02 2.09 4.06 2.43

ALEPH 27 0.41 0.68 0.48 1.09 0.59 0.35

DELPHI 28 2.96 2.59 3.42 5.69 3.75 1.80

L3 28 2.16 1.96 1.52 3.62 2.03 0.94

OPAL 26 1.54 2.03 1.79 3.91 1.83 1.02

All 170/166/162 1.91 1.93 2.08 2.83 2.03 1.16

Q2 > 4 GeV2 137 1.42 1.56 1.48 2.72 1.66 1.13

Table 6: Quantitative comparison of predictions of selected parameterizations with measure-ments of F γ

2 . Listed are the number of data points (ndf) and the values of χ2/ndf calculatedusing Eq. 83 for the individual experiments, for all data points (All) and for the measure-ments with Q2 > 4 GeV2. The number of data points used in the comparison depends on thelowest value of Q2 for which a parameterisation of F γ

2 is valid. The TPC/2γ measurementat 〈Q2〉 = 0.24 GeV2 is outside the validity of all the parameterisations listed here. Theminimum Q2 for the GRSc parameterisation is Q2

0 = 0.5 GeV2 and for the CJKL param-eterisation is Q2

0 = 1 GeV2. This reduces the number of TPC/2γ data points used in thecomparison to 15 and 11, for GRSc and CJKL parameterisations, respectively.

with χ2/ndf of 0.69 and 0.13, respectively. The results of the fit are in agreement with therecent L3 measurement [146] of F γ

2 at Q2 = 120 GeV2. There is also a nice agreementbetween OPAL and L3 measurements.

8.2.4 Charm structure function of the photon

The first measurement of the photon structure function F γ2 for heavy quarks has been per-

formed by the OPAL experiment in [141]. This measurement has been later extended tothe full LEP statistics [142]. The production of charm quarks has been studied in deepinelastic electron-photon scattering using data collected at e+e− centre-of-mass energies√

see = 183 − 209 GeV, with a luminosity weighted average of√

see = 196.6 GeV. Thecharm quarks have been identified by full reconstruction of charged D⋆ mesons using theirdecays into D0π with D0 observed in two decay modes.with charged particle final states, Kπand Kπππ. Fig. 38 shows the difference between the D⋆ and D0 candidate masses for bothdecay channels combined. A clear peak is observed around the mass difference between theD⋆ and the D0 meson, which is ∆m0 = 0.1454 GeV [1]. In Fig. 38 we also show a fit of aGaussian for the signal and a power law function of the form a(∆m − mπ)b, for the back-

66

0

0.2

0.4 TPC/2γ (0.24)TPC/2γ (0.38)TPC/2γ (0.71)

Fγ 2

/α TPC/2γ (1.3)L3 (1.9)OPAL (1.9)

TPC/2γ (2.8)PLUTO (2.4)

GRV HO SaS 1D

0

0.5

PLUTO (4.3)L3 (5.)OPAL (3.7)

TPC/2γ (5.1)TOPAZ (5.1)AMY (6.8)

DELPHI (5.2)

PLUTO (9.2)ALEPH (9.9)OPAL (8.9)

0

0.5

L3 (10.8)DELPHI (12)DELPHI (12.7)

OPAL (10.2)

TOPAZ (16)L3 (15.3)ALEPH (17.3)

OPAL (17.8)

ALEPH (20.7)DELPHI (19)OPAL (20)

0

0.5

1 TASSO (23)JADE (24)L3 (23.1)

DELPHI (28.5)

PLUTO (45)DELPHI (40)

TOPAZ (80)AMY (73)ALEPH (67.2)

0

0.5

1

10-3

10-2

10-1

1

JADE (100)L3 (120)

DELPHI (101)

x

AMY (390)ALEPH (284)

DELPHI (700)

OPAL (780)

Figure 36: Compilation of all available measurements of F γ2 as a function of x. The measured

values of F γ2 /α are compared to predictions from the GRV HO and SaS1D parameterisa-

tions of the parton distribution functions. The curves are drawn for the average Q2 of themeasurements shown in a particular plot. The statistical error is given by the inner errorbars, and the statistical and systematic errors added in quadrature (or the total error) asfull error bars. The numbers in parentheses close to the name of experiment denote average〈Q2〉 of the measurement.

ground contribution. An absolute prediction of the combinatorial background measured fromthe data using events with a wrong charge pion for the D⋆ → D0π decays. The cross sectionσD⋆

for production of charged D⋆ in the reaction e+e− → e+e−D⋆X has been measured in arestricted kinematical region where pD⋆

t > 1 GeV for an electron scattering angle of 33–55mrad, or pD⋆

t > 3 GeV for 60–120 mrad, |ηD⋆| < 1.5 and 5 < Q2 < 100 GeV2 in two bins of

67

0

2

4

6

8

10

12

10-1

1 10 102

103

Q2 [GeV2]

Fγ 2,

QE

D /α

+ N

TPC/2γ

TOPAZ

TASSO

PLUTO

AMY

JADE

L3

ALEPH

DELPHI

OPAL

0.005 1

0.05 2

0.15 3

0.25 4

0.35 5

0.50 6

0.65 7

0.75 8

0.90 9

GRV HO SaS 1D

<x> N

Figure 37: Compilation of all available measurements F γ2 as a function of Q2 in several bins

of the variable x. The measured values of F γ2 /α are compared to the predictions based on the

GRV HO and SaS1D parameterisations of the parton distribution functions. The statisticalerror is given by the inner error bars, and the statistical and systematic errors added inquadrature (or the total error) as full error bars. The bin boundaries in x are 0.001, 0.01,0.1, 0.2, 0.3, 0.4, 0.6, 0.8, 0.99, with central values 〈x〉 indicated in the figure. To separatethe measurements in different bins of x, an offset N shown in the figure has been added toeach measurement and theoretical prediction.

x, 0.0014 < x < 0.1 and 0.1 < x < 0.87. Within errors the cross section can be describedby the HERWIG model. From σD⋆

the charm production cross section σ(e+e− → e+e−ccX)and the charm structure function of the photon F γ

2,c are obtained by extrapolation in thesame bins of x and Q2, using Monte Carlo. The results of the measurement are presented inFig. 39 and listed in Table 26. For x > 0.1, the perturbative QCD calculation at NLO agreesperfectly with the measurement. For x < 0.1 the point-like component, however calculated,lies below the data. Subtracting the NLO point-like prediction, a measured value for thehadron-like part of 0.154 ± 0.059 ± 0.029 is obtained.

68

0

5

10

15

20

25

30

35

40

45

50

0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21

OPAL

fit: 55.3 ± 11.0 signal events

wrong-charge combination

∆m [GeV]

Eve

nts

0

20

40

60

80

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x

σ(e+ e- →

e+ e- cc- X

) [p

b] a)OPALNLO (Laenen et al.)LOHERWIG

0

0.1

0.2

0.3

0.4

10-3

10-2

10-1

1x

Fγ 2,c (

x,Q

2 ) / α OPAL

NLO (Laenen et al.)NLO hadron-likeLO

b)GRS-LOpoint-like

Q2 = 20 GeV2

Figure 38: Distribution of the difference, ∆m, between the D⋆ and D0 candidate masses forboth decay channels combined. The data are shown as points with statistical errors. Thehistogram represents the combinatorial background estimated using events with a wrongcharge pion for the D⋆ → D0π decays. The curve is the result of the fit to a Gaussian signalplus a power-law background function, as explained in the text.

Figure 39: OPAL results for (a) the cross section σ(e+e− → e+e− cc X), with 5 < Q2 < 100GeV2 and (b) for the charm structure function of the photon divided by the fine structureconstant, F γ

2,c(x, Q2)/α, at Q2 = 20 GeV2. The data points are the results obtained withthe HERWIG Monte Carlo model. The outer error bar is the total error and the inner errorbar the statistical error. The data points in (b) are placed at those x values that correspondto the average predicted F γ

2,c within a bin. The data are compared to the calculation of [26]performed in LO and NLO. The band for the NLO calculation indicates the theoretical errorfrom uncertainties in the charm quark mass and renormalisation and factorization scales. In(a) the cross section prediction of the HERWIG Monte Carlo model is also given; (b) alsoshows the prediction of the GRS-LO parameterisation for the structure function at Q2 = 20GeV2 and its point-like component separately.

8.3 Hadronic structure function of the electron

The measurements of the hadronic structure function of the electron F e2 have been performed

so far only by the OPAL [143] and DELPHI [144] experiments. The OPAL analysis has beenperformed by the author of this paper. In the following we present the results obtained byboth experiments and compare them with theoretical predictions.

The OPAL analysis is essentially an extension of the analysis of the photon structurefunction F γ

2 presented in Section 8.2.1. In particular, exactly the same data and MonteCarlo samples have been used in both analyses. Therefore for the list of selection cuts, thediscussion of the background processes and the quality of description of the data by theMonte Carlo models the reader is referred to Section 8.2.1. Here only some additional infor-mation on the differences between the measurements of F γ

2 and F e2 will be given.

69

Phase space for the measurement of dσee/dz

OPAL DELPHI

E ′ > 0.75Eb E ′ > 0.3Eb

34 < θ < 55 mrad 35 < θ < 155 mrad

W >√

3 GeV W > 3 GeV

Table 7: Phase space used for the measurements of the cross section dσee/dz for the processe+e− → e+e− hadrons, by OPAL and DELPHI experiments.

The variable z, in opposite to the variable x, is calculated with respect to the beamparticle (electron) of fixed and known momentum. It is informative to see the distribution ofsingle tagged events in the kinematical plane (z, Q2). As an example we show in Fig. 40 thedistribution of single tagged events, selected in the year 1999, in the (z, Q2) plane togetherwith several lines representing fixed values of different kinematical variables.

One of the arguments in favour of measuring the electron structure function F e2 is a much

better reconstruction of the variable z than the variable x. This can be seen in Fig. 41 wherethe migrations of events in the (x,Q2) and (z,Q2) planes as well as the smearings of the vari-ables x and z are shown. The plots in Fig. 41 are based on the HERWIG model. The similarplots obtained from PHOJET show the same general features. The base of an arrow is at theaverage generated values of x and Q2 (z and Q2) for a given bin. The head of the arrow is atthe average reconstructed x and Q2 (z and Q2). The migration between different Q2 bins issmall. Also the migration between different z bins is rather small. However, the migrationbetween different x bins is very large, and any attempt to measure x distribution requires anunfolding method. The same feature can be seen on the plots showing the smearings of thevariables x and z. The variable z is fully determined by the energy and polar angle of thescattered beam electron, which are usually well reconstructed, whereas the measurement ofthe variable x requires additional information on the hadronic final state, which is not fullymeasured nor well modeled.

Due to small migrations in the variable z the measurement of the differential cross sec-tion dσee/dz, does not require an unfolding procedure and a simple bin-by-bin correction issufficient. As one can see from Fig. 40 the phase space available for the measurement is noteasily divisible into rectangle bins in z and Q2 unless one chooses very small bins (especiallyin small z region) or decides to fulfill the missing phase space regions using a model. Thestatistics does not allow to choose sufficiently small bins. As we do not want to bias themeasurement assuming any model we decided to measure the cross section as a function of zin the phase space region defined by the cuts imposed by the experimental conditions. Theyare listed in Table 7. The bins in z have been chosen to be equidistant in the logarithmicscale and are given in Table 9.

The measured cross section dσee/dz is further corrected for the QED radiative effects,which were estimated using the TWOGAM 2.04 program, in a similar way as for the measure-ment of F γ

2 . The calculations are performed using mixed variables, which means that z andQ2 are calculated from electron variables, while W 2 is calculated from hadronic variables.Samples of TWOGAM with radiative corrections switched on and off corresponding to the

70

z

Q2 [G

eV2 ]

0

5

10

15

20

25

30

35

10-4

10-3

10-2

10-1

Figure 40: Kinematical plane (z,Q2) with single tagged events selected by the OPAL ex-periment in the year 1999. Several lines representing fixed values of different kinematicalvariables are plotted. The beam energy used for the lines is Eb = 99 GeV.

Figure 41: Migration of events in the (x,Q2) and (z,Q2) planes is shown on the upper plotsfor single tagged events using HERWIG Monte Carlo model. The base of an arrow is at theaverage generated x and Q2 (z and Q2) for a given bin. The head of the arrow is at theaverage reconstructed x and Q2 (z and Q2). Smearing of the variables x and z is shown onthe bottom plots.

z

(σbo

rn-σ

rad)

/σra

d

full qpm qcd vmd

z

rel.

frac

. [%

]

-20

-10

0

10

20

30

10-3

10-2

10-1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

10-3

10-2

10-1

Figure 42: QED radiative corrections estimated with TWOGAM 2.04 Monte Carlo programin the phase space used for F e

2 measurement by OPAL [143]. The relative difference of non-radiative and radiative with respect to the radiative cross section is shown for 〈Q2〉 = 15GeV2 as a function of z for different subprocesses generated by TWOGAM as well as for theweighted average. The relative size of the individual subprocesses is shown in bottom plot.

71

three average centre-of-mass energies of the data were generated separately. To each samplethe experimental restrictions on the electron energy and polar angle, the minimum and max-imum invariant mass and the anti-tag energy and angle were applied. The predicted ratio ofthe differential cross section for each bin of the analysis is used to correct the data, i.e. themeasured differential cross section dσee/dz is multiplied by the ratio of the non-radiative tothe radiative cross sections predicted by TWOGAM. The radiative corrections with statisti-cal errors, weighted with the luminosity of the data samples, are given in Table 8. In Fig. 42we show the value of the radiative corrections as a function of z for different subprocessesgenerated by TWOGAM as well as for the weighted average. Also the relative contributionof the individual subprocesses to the total cross section is shown in the bottom plot.

The electron structure function F e2 was obtained from the differential cross section

dσee/dz dQ2 using Eq. 69. Although it is reasonable to measure the cross section in thephase space defined by the cuts on the scattered electron polar angle and its energy, thestructure function F e

2 should be given as a function of z for an average value of Q2. Thereforethe structure function obtained from the cross section is further corrected to the bin-centrein the variable z and to the nominal value of Q2 = 15 GeV2, which is close to the averageQ2 in the high z range. The overal bin-centre correction is calculated as the average ofGRV LO and SaS1D predictions for the correction from the average F e

2 over the bin in zand the range of Q2 to the value of F e

2 at the nominal (x,〈Q2〉 =15 GeV2) position. Theluminosity weighted average values of bin-centre corrections and the nominal values of z and〈Q2〉 are given in Table 8. The errors on the bin-centre correction are obtained as a halfof the difference between the predictions of GRV LO and SaS1D parameterisations. Themeasured electron structure function F e

2 and the cross section dσee/dz do not need to becorrected for the P 2 effect (as in case of F γ

2 ). However, the measured F e2 should be compared

with predictions which involve a dependence of F γ2 on P 2. Most of parameterisations of F γ

2

available are only valid for P 2 = 0. Therefore all predictions used for the comparisons withmeasured F e

2 were corrected for the P 2 effect. The correction factor is defined as the ratio ofF e

2 (z, Q2) obtained from the convolution of F γ2 (x, Q2, P 2) and fγ⋆/e(y, P 2) (see Eq. 70), and

of F e2 (z, Q2) obtained from the convolution of F γ

2 (x, Q2, P 2 = 0) and fγ⋆/e(y, P 2).

radiative bin-centrez range

cor. [%] cor. [%]

0.0009 - 0.00265 −12.62 ± 0.24 8.08 ± 0.53

0.00265 - 0.0078 −5.79 ± 0.21 −2.23 ± 0.05

0.0078 - 0.0230 0.82 ± 0.26 −4.22 ± 0.05

0.0230 - 0.0677 10.54 ± 0.34 −4.74 ± 0.09

0.0677 - 0.2000 20.78 ± 0.51 −4.41 ± 0.15

0.2000 - 0.9048 27.53 ± 0.95 2.59 ± 0.28

Table 8: The average values of the corrections which have been applied are listed. For agiven bin in z the correction is the difference of the corrected and non corrected cross sectionsas a percentage of the non corrected cross section. For the details of how the correction havebeen estimated and the meaning of the errors see text.

72

z

F2e (z

)/α2

⟨Q2⟩=15 GeV2

LAC1

CJKL

GRSc LO

GRV LO

WHIT1

SaS1D

QPM

OPAL

0

2

4

6

8

10

12

14

16

18

20

10-3

10-2

10-1

Figure 43: Electron structure function F e2 divided by the fine structure constant squared

shown as a function of z for an average 〈Q2〉 = 15 GeV2. The points represent the OPALdata with statistical (inner error bars) and total (outer error bars) errors. Predictions ofseveral parameterisations are shown as curves.

The measured differential cross section dσee/dz for the process e+e− → e+e−γ⋆γ → e+e−Xin the phase space region defined in Table 7 and the measured structure function F e

2 foraverage 〈Q2〉 = 15 GeV2 are listed in Table 9 together with statistical and systematic errors.The systematic errors were estimated using essentially the same list of systematic checksas in the case of the measurement of F γ

2 described in Section 8.2.1 (the systematic errordue to the P 2 effect and due to the check (8) are not applicable in the present analysis).The contribution from individual checks to the total systematic error are given in Table 10as a percentage of the nominal value of F e

2 . The most significant contribution to the totalsystematic error comes from the energy scale uncertainty in the determination of the en-ergy of the tagged electron. The effect of the modeling of the hadronic final state is onlysignificant in the highest z bin. The electron structure function F e

2 measured by the OPALexperiment is shown in Fig. 43 and is listed in addition in Table 27. The data are comparedto the predictions based on several parameterisations of F γ

2 convoluted with the photon fluxaccording to Eq. 70. All theoretical predictions shown in Fig. 43 were corrected for the P 2

73

z

range bin-centredσ/dz [pb] F e

2 /α2

0.0009 - 0.00265 0.00154 6128 ± 57 +518−485 13.10 ± 0.12 +1.12

−1.04

0.00265 - 0.0078 0.00455 2992 ± 22 +280−236 10.04 ± 0.08 +1.22

−0.79

0.0078 - 0.0230 0.01339 857 ± 8 +28−47 7.84 ± 0.07 +0.26

−0.43

0.0230 - 0.0677 0.03946 220 ± 3 +30−28 5.76 ± 0.09 +0.76

−0.73

0.0677 - 0.2000 0.11636 48 ± 1 +8−7 3.66 ± 0.10 +0.64

−0.55

0.2000 - 0.9048 0.42539 2.8 ± 0.2 +1.0−0.6 0.71 ± 0.03 +0.25

−0.13

Table 9: Results for the cross section dσ/dz obtaineded in the phase space defined byE ′ > 0.75Eb, 34 < θ < 55 mrad and W >

√3 GeV and the structure function F e

2 /α2 forthe average 〈Q2〉 = 15 GeV2 measured by the OPAL experiment [143]. The first errors arestatistical and the second systematic.

effect using average prediction of SaS1D and GRSc parameterisations of F γ2 for the virtual

photon and the method described above. The data clearly exclude the quark parton model(QPM). Also the rise as fast as predicted by the LAC1 parameterisation is disfavoured bythe data. However, most of the modern predictions are in accord with the measured F e

2 .The electron structure function F e

2 has been also measured at LEP by the DELPHI exper-iment [144]. The general procedure of extracting F e

2 from the single tagged events is similarto that applied in OPAL. The data samples used in the analysis correspond to integratedluminosities of 72 pb−1 accumulated by the DELPHI experiment at e+e− centre-of-mass en-ergy

√see = 92.5 GeV (LEP1), and of 487 pb−1 accumulated at several e+e− centre-of-mass

energies in the range√

see = 189 − 209 GeV (LEP2).Single tagged events are selected by applying cuts on the scattered electrons and on the

hadronic final state. A scattered electron is required to have energy E ′ > 0.6Eb and polarangle in the range 43.6 < θ < 174.5 mrad. This corresponds to the effective ranges of photonvirtualities of 4.5 < Q2 < 40 GeV2 at LEP1 and 16 < Q2 < 80 GeV2 at LEP2. To ensurethat the virtuality of the quasi-real photon is small, the highest energy cluster in the hemi-sphere opposite to the one containing the tagged electron must have an energy Ea < 0.3Eb.The visible invariant mass Wvis is required to be greater than 3 GeV, and at least 3 tracksshould be present in the hadronic final state. After these cuts, 2168 single tagged eventswere selected at LEP1, and 9697 events at LEP2 energies. Among them we expect 111 atLEP1 and 1027 at LEP2 background events, coming mainly from two photon production oftau pairs, four fermion final states and hadronic decays of Z0 bosons. The data are correctedfor detector effects using TWOGAM 2.03 Monte Carlo model.The electron structure function F e

2 is obtained from the differential cross section dσee/dzdQ2

in the similar way as in the OPAL analysis. The cross section is measured in the phase spacedefined in Table 7. The electron structure function F e

2 is measured in several bins of Q2,which effectively represent stronger cuts on the available phase space than the cuts listedin Table 7. The following ranges of Q2 are choosen: 4.5 < Q2 < 40 GeV2 at LEP1 en-ergy, and four intervals at LEP2 energies: 16 < Q2 < 20 GeV2, 20 < Q2 < 30 GeV2,30 < Q2 < 50 GeV2 and 50 < Q2 < 80 GeV2. The radiative corrections have been estimated

74

using TWOGAM 2.04 program. The size of the corrections and the estimation method aresimilar to that used by OPAL. The bin centre corrections are below 1% and were not appliedto the data. The structure function F e

2 measured by the DELPHI experiment at LEP1 islisted in Table 28 and the one measured at LEP2 is shown in Fig. 44 and listed in Table 29.In Fig. 44 the data are compared to several theoretical predictions of F e

2 . For details seethe caption of the figure. As in case of the OPAL measurement the measured F e

2 clearlydisfavour QPM and steeply rising predictions like LAC1 (not shown), and due to large errorshave no sufficient power to distinguish between the predictions shown in the figure.

75

z bin total error

centreF e

2 /α2

stat. sys.θmin− θmin+ ESW+ ESW− Ehad+ Ehad− Wmin− Wmin+ Wmax+ Wmax−

0.00154 13.10 0.9 +8.5−7.9 1.52 −1.44 −7.00 7.87 0.06 0.03 1.13 −1.41 0.45 −0.28

0.00455 10.04 0.8 +12.2−7.9 0.57 −0.62 −6.46 8.90 0.21 −0.27 2.15 −2.27 0.05 −0.06

0.01339 7.84 0.9 +3.3−5.5 0.30 −0.36 −2.66 −1.58 0.29 −0.25 2.50 −2.39 0.00 −0.02

0.03946 5.76 1.6 +13.2−12.7 0.37 −0.54 12.42 −11.73 0.48 −0.23 3.11 −2.35 0.00 0.00

0.11636 3.66 2.7 +17.5−15.0 0.26 −0.48 10.58 −4.36 0.59 0.00 2.85 −1.55 0.00 0.00

0.42539 0.71 4.2 +35.2−18.3 0.54 −0.43 21.85 3.90 0.22 −0.39 0.22 −4.71 0.00 0.00

z bin total error MC bin rad.

centreF e

2 /α2

stat. sys. model centre corr. Nch > 3 |z0|− |z0|+ |d0|− |d0|+

0.00154 13.10 0.9 +8.5−7.9 0.75 0.53 0.24 −1.39 −2.82 1.88 −0.51 0.45

0.00455 10.04 0.8 +12.2−7.9 0.92 0.05 0.21 −2.73 −2.91 1.69 −0.52 0.27

0.01339 7.84 0.9 +3.3−5.5 0.89 0.05 0.26 −1.64 −3.08 1.77 −0.63 0.41

0.03946 5.76 1.6 +13.2−12.7 1.25 0.09 0.34 0.38 −3.40 1.58 −0.81 0.07

0.11636 3.66 2.7 +17.5−15.0 2.10 0.15 0.51 −1.36 −4.35 3.00 −0.92 0.56

0.42539 0.71 4.2 +35.2−18.3 11.64 0.28 0.95 2.63 −7.33 3.79 −0.64 0.00

Tab

le10:

System

aticerrors

forF

e2/α

2as

afu

nction

ofz

forth

eaverage

〈Q2〉

=15

GeV

2.T

he

statisticalerror

and

the

contrib

ution

sto

the

system

aticerror

fromeach

ofth

esou

rcesare

listedas

ap

ercentages

ofF

e2/α

2.

76

0

5

10

15

20

25

-2.8 -2.7 -2.6 -2.5 -2.4 -2.3

GRV-LO-INT-P1

SaS1D-INT-P1

GRV-HO-INT-P19

SaS1D-INT-P19

0

2

4

6

8

10

12

14

16

18

-2.5 -2.25 -2 -1.75 -1.5 -1.25 -1 -0.75 -0.5

GRV-LO-INT-P1

SaS1D-INT-P1

GRV-HO-INT-P19

SaS1D-INT-P19

SaS1D-INT-PP19

0

2

4

6

8

10

12

14

16

18

-2.25 -2 -1.75 -1.5 -1.25 -1 -0.75 -0.5 -0.25 0

GRV-LO-INT-P1

SaS1D-INT-P1

GRV-HO-INT-P19

SaS1D-INT-P19

SaS1D-INT-PP19

0

2

4

6

8

10

12

14

16

18

-2 -1.75 -1.5 -1.25 -1 -0.75 -0.5 -0.25 0

GRV-LO-INT-P1

SaS1D-INT-P1

GRV-HO-INT-P19

SaS1D-INT-P19

SaS1D-INT-PP19

Figure 44: Electron structure function F e2 divided by the fine structure constant squared is

shown as a function of z in four intervals of Q2 displayed together with the average valueson the plots. The points represent the DELPHI LEP2 data with statistical (inner errorbars) and total (outer error bars) errors. Predictions of several parameterisations are shownas curves. The predictions are obtained from the convolution of F γ

2 and the photon fluxfγ⋆/e(y, P 2) as given in Eq. 70. The integration over P 2 extends to 1 GeV2 (INT-P1) or to19 GeV2 (INT-P19) in case when P 2 dependence is taken into account only in the photonflux, and to 19 GeV2 (INT-PP19) in case of SaS1D parameterisation with P 2 dependencetaken into account in both photon flux and F γ

2 .

77

9 Interactions of two virtual photons

In this Section we present the results on the structure of the highly virtual photon and onthe structure of interactions of two highly virtual photons. The effective structure functionof the virtual photon F γ

eff can be measured in the kinematical region where Q2 ≫ P 2 & Λ.If both photons have similar virtualities then the interference terms in the cross section(Eq. 13) can be large and there is no clear relation between the photon structure functionsand the individual cross sections. In this situation the most proper measurements are thedifferential e+e− or total γ⋆γ⋆ cross sections. The effective structure function F γ

eff has beenmeasured so far only for the hadronic final state by the PLUTO and L3 experiments. Crosssections for the interactions of virtual photons have been measured for both leptonic (OPAL)and hadronic (OPAL, L3, ALEPH) final states.

9.1 Leptonic cross section for the scattering of two virtual photons

The differential cross section for the reaction e+e− → e+e− µ+µ−, proceeding via the ex-change of two highly virtual photons has been measured at LEP by the OPAL experi-ment [111]. The results of the measurement are shown in Fig. 45 and listed in Table 19. Themeasurements have been performed in two ranges of probe and target photon virtualities:1.5 < Q2 < 6 GeV2, 1.5 < P 2 < 6 GeV2 and 5 < Q2 < 30 GeV2, 1.5 < P 2 < 20 GeV2.The data are compared with the predictions from the Vermaseren and GALUGA programs.The data are well described using the full cross section from Eq. 13. With the GALUGAprogram, it is possible to see the influence of the non-vanishing terms proportional to cos φand cos 2φ. If the terms are neglected, the predicted cross section grossly overestimates themeasured cross section. This shows that both terms, τTT and τTL, are present, mainly atx > 0.1, and that their contributions to the cross section are negative.

9.2 Effective hadronic structure function of virtual photon

The first measurement of the effective structure function of the virtual photon F γeff has been

performed by the PLUTO experiment [145]. The results of this measurement are presentedin Fig. 46 and listed in Table 30. The average virtuality of the probe photons was 〈Q2〉 = 5GeV2 and of the target photons 〈P 2〉 = 0.35 GeV2. In Fig. 46a the measured structurefunction F γ

eff as a function of x is compared with the predictions of F γeff from the double

virtual box diagram and with the predictions of the parameterizations GRSc and SaS1Dfor F γ

2 and for (F γ2 + 3/2F γ

L), with F γL calculated in all cases from the box diagram. In

Fig. 46b the P 2 evolution of F γeff measured by PLUTO is compared to the QPM predictions

and to the predictions of the parameterizations GRSc and SaS 1D for (F γ2 + 3/2F γ

L), withF γ

L calculated in all cases using the box diagram.The only measurement of F γ

eff at LEP has been performed by the L3 experiment [146].The results of this measurement are shown in Fig. 47 and listed in Table 31. The average

78

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x

dσ /

dx [p

b]

(a)

< Q2> = 3.6 GeV2

< P2> = 2.3 GeV2

OPALVermaserenGalugaτTT = 0τTT,τTL = 0

x

dσ /

dx [p

b]

(b)

< Q2> = 14.0 GeV2

< P2> = 5.0 GeV2

OPALVermaserenGalugaτTT = 0τTT,τTL = 0

0

5

10

15

20

25

30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 45: The measured differential cross sections dσ/dx for two sets of average valuesof 〈Q2〉 = 3.6 GeV2, 〈P 2〉 = 2.3 GeV2 and 〈Q2〉 = 14 GeV2, 〈P 2〉 = 5 GeV2. The pointsrepresent the OPAL data with their statistical (inner error bars) and total errors (outererror bars). The full line denotes the differential cross section as predicted by the VermaserenMonte Carlo using the same bins as for data. The additional three histograms represent thecross section calculations from Galuga Monte Carlo for three different scenarios: the fullcross section (solid line), the cross section obtained for vanishing τTT (dot-dash) and thecross section obtained for vanishing τTT and τTL (dash). The tick marks at the top of thefigures indicate the bin boundaries of the OPAL measurement.

virtualities of the photons are 〈Q2〉 = 120 GeV2 and 〈P 2〉 = 3.7 GeV2. In Fig. 47a themeasured structure function F γ

eff as a function of x is compared with the predictions of F γeff

from the double virtual box diagram and with the predictions of the parameterizations GRScand SaS1D for F γ

2 and for (F γ2 + 3/2F γ

L). In Fig. 47b the P 2 evolution of F γeff measured

by L3 is compared to the QPM predictions and to the predictions of the parameterizationsGRSc and SaS1D for (F γ

2 + 3/2F γL). In all cases the longitudinal structure function F γ

L iscalculated using the box diagram approximation.

Within the present statistics, the measurements of F γeff are in full agreement with the

naive QPM predictions derived from the doubly virtual box diagram. Only the point atlowest x in the PLUTO measurement may be a sign of a QCD resumed parton content inthe virtual photon. However, to firmly establish this, measurements at still lower values ofx and with better experimental precision are necessary. The measurements lie above thepredictions of F γ

2 demonstrating that the effectively measured quantity is (F γ2 +3/2F γ

L).

79

x

Fγ eff /

α

PLUTO ⟨Q2⟩ = 5 GeV2

⟨P2⟩ = 0.35 GeV2(a)

Fγeff / α

GRScSaS 1D

(F2γ+3/2FL

γ) / α F2γ / α

P2 [GeV2]

Fef

f / α

PLUTO ⟨Q2⟩ = 5 GeV2 (b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1

x

Fγ eff /

α

L3 ⟨Q2⟩ = 120 GeV2

⟨P2⟩ = 3.70 GeV2(a)

Fγeff / α

GRScSaS 1D

(F2γ+3/2FL

γ) / α F2γ / α

P2 [GeV2]

Fef

f / α

(b)L3 ⟨Q2⟩ = 120 GeV2

00.20.40.60.8

11.21.41.61.8

22.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8

Figure 46: The effective structure function of the virtual photon F γeff divided by the fine

structure constant measured by the PLUTO experiment [145] is shown as a function ofx (a) and as a function of P 2. Points represent the data with total errors. In (a) thecurves are the predictions for F γ

eff from the doubly virtual box diagram for massless partons(solid), and the predictions for F γ

2 and for (F γ2 +3/2F γ

L) from GRSc (dash) and SaS1D (dot)parameterisations, with F γ

L always calculated from the box diagram. In (b) the curves arethe predictions for F γ

eff (solid) and for (F γ2 +3/2F γ

L) from GRSc (dash) and SaS 1D (dot)parameterisations, with box diagram calculations for massive quarks.

Figure 47: The effective structure function of the virtual photon F γeff divided by the fine

structure constant measured by the L3 experiment [146] is shown as a function of x (a) andas a function of P 2. The meaning of curves is the same as in Fig. 46.

The P 2 dependence of the PLUTO measurement suggests, as expected, a slow decrease withincreasing P 2, but it is also consistent with a constant behaviour. The L3 measurement iseven less conclusive.

9.3 Hadronic cross section for the γ⋆γ⋆ scattering process

The hadronic structure of interactions of virtual photons has been studied, using the reactione+e− → e+e− hadrons, by three LEP experiments: OPAL [149], L3 [150, 151] and ALEPH[152]. The measurement of the hadronic cross section for interactions of two virtual photonsat OPAL has been performed by the author of this paper. The measurements are basedon double tagged events, where both scattered electrons are detected mostly in the smallangle calorimeters. The first published measurement of this type has been performed by theL3 experiment [150] at e+e− centre-of-mass energies

√see = 91 GeV and

√see = 183 GeV.

80

However, the data in this analysis had been not corrected for QED radiative effects, whichwere found later to be very large. Therefore we do not discuss this first L3 measurementin the following. The next measurement by L3 as well as the measurements by OPAL andALEPH have been performed at LEP2 energies,

√see = 189 − 209 GeV, with a luminosity

weighted average of around√

see = 198 GeV. The total luminosities used by the experimentsamount to 592.9 pb−1 (OPAL), 640 pb−1 (ALEPH) and 617 pb−1 (L3).

9.3.1 Event selection and background estimation

Double tagged two photon events were selected with the sets of cuts discussed below andwith the values of the cuts specified in Table 11. In each experiment two electron candidateswith sufficient energies Ei and within the specified range of polar angles θi must be observed.To ensure that the final state is a hadronic system and not a pair of leptons, at least threetracks (Nch) are required by OPAL and ALEPH, or five objects (Npart) defined as tracksor isolated calorimeter clusters, by L3. In each analysis also a minimum visible invariantmass Wvis is required. It is reconstructed from tracks measured in the central tracking detec-tors and the position and energy of clusters measured in the electromagnetic and hadroniccalorimeters. In order to ensure that the event is well contained in the detector and to reducebackground from beam–gas interactions, it is required by OPAL that the z component of thetotal momentum vector of the event, |

∑

pz|, is less than 35 GeV, and that the total energymeasured in the event is less than 2.2Eb. For the background reduction ALEPH requirs thetotal visible energy to be above 1.4Eb.The following additional cuts were used only by OPAL. In order to restrict the number ofelectron candidates to exactly two, there should not be observed any additional single objectwith energy above 0.25Eb. To further suppress the beam-gas background a strict require-ments on the vertex position in both z direction and in the plane perpendicular to the z-axis,have been set. Here 〈z0〉 is calculated as the error weighted average of the z coordinates ofall tracks at the point of closest approach to the origin in the (r, φ) plane, and d0 is thedistance of the primary vertex from the beam axis.Remaining Bhabha-like events (i.e. a Bhabha event with a random overlap of hadronicactivity) were tagged in all experiments using the back-to-back topology of the scatteredelectrons and removed from the data samples.

With these cuts 179, 491 and 891 events were selected by OPAL, L3 and ALEPH, re-spectively. Among them we expect background events from two main sources. The firstcontribution is from e+e− processes containing scattered beam electrons in the final stateobserved in the detector, and the second stems from coincidences of e+e− processes withoutthe electrons in the final state and with off-momentum electrons from beam–gas interactionsfaking the scattered electrons.

The background from e+e− reactions containing electrons in the final state, dominantlystems from the processes e+e− → e+e− τ+τ− and e+e− → e+e− e+e−, and was estimatedusing the Vermaseren [73] (OPAL, L3) or PHOT02 [60] (ALEPH) Monte Carlo programs.The contribution from other e+e− backgrounds, such as e+e− → e+e− qq and other pro-cesses leading to four fermion final states was found to be much less important. L3 reporteda significant background from single-tagged two-photon processes with a cluster from the

81

OPAL ALEPH L3

Ei > 0.4Eb Ei > 0.3Eb Ei > 40 GeV34 < θi < 55 mrad 35 < θi < 55 mrad 30 < θi < 66 mrad

60 < θi < 155 mradNch ≥ 3 Nch ≥ 3 Npart ≥ 5Wvis > 5 GeV Wvis > 3 GeV Wvis > 2.5 GeVEmax < 0.25Eb

|z0| < 4 cm|d0| < 0.5 cm12

∑

E < 1.1Eb12

∑

E > 0.7Eb

|∑

pz| < 35 GeVrejection of Bhabha–like events

Table 11: Selection cuts for double tagged events. For description of the variables see text.

hadronic final state misidentified as the second electron. This type of background was negli-gible in OPAL and ALEPH. The reason might be that the acceptance of the L3 small anglecalorimeters start from a significantly lower polar angle than in the other two experiments.

Beam–gas interactions can result in off-momentum electrons observed in the detectors,faking final state electrons from the process e+e− → e+e− hadrons. Due to the LEP runningconditions this kind of background was high in OPAL and ALEPH experiments and onlynegligible in case of L3. The details of the background estimation in OPAL are given below.A similar procedure was adopted by ALEPH. The background was estimated using a sam-ple of Bhabha events, selected by requiring events with two back-to-back electrons in theSW calorimeters, which each have an energy of more than 0.7Eb, with |∆r| < 0.5 cm and(π − 0.1) < |∆φ| < π rad, where ∆r and ∆φ are the differences of the radial and azimuthalcoordinates of the tagged electrons. Additional clusters in the SW detectors, which fulfill thecriteria for electron candidates, but do not belong to the Bhabha event, are counted as off-momentum electrons. The probabilities to have an overlapping off-momentum electron withan event coming from the interaction region are determined for the left (−z) / right (+z)side of the detector separately and amount to 0.000715/0.00115 (1998), 0.00139 /0.00279(1999) and 0.00117 /0.000671 (2000). The relative statistical precision of these probabilitiesis 2-3%. As the running conditions changed from year to year also the off-momentum back-ground was changing accordingly. The background expectation is more than a factor twolarger in 1999 than in the other years. Using the above overlap probabilities, and assumingthat they are independent between the left and right side of the detector, we predict thatthere should be 14.6 events with ’double’ overlaps in the Bhabha sample, which agrees wellwith the 19 events observed.

It is essential to check this method of estimating the background from off-momentumelectrons on a different process. Here we used a sample of single-tagged two-photon events.The sample is selected with the same cuts as described for the double-tagged selection, ex-cept that only one scattered electron is required. The event properties are compared with theabsolute prediction for the single-tagged plus background events. The single-tagged eventsare based on the HERWIG generator with the GRV [74] parameterisations of the photonstructure function F γ

2 , which has been shown to describe the single-tagged cross-section [75]

82

OPALHERWIG+bgr.off-mom. bgr.physics bgr.

a)

EL/Ebev

ents

1999 b)

ER/Eb

even

ts

c)

φL [rad]

even

tsd)

φR [rad]

even

ts

e)

θL [rad]

even

ts

f)

θR [rad]

even

ts

0

500

1000

0.4 0.6 0.8 10

250

500

750

1000

0.4 0.6 0.8 1

0

250

500

750

1000

0 2 4 60

500

1000

1500

0 2 4 6

0

500

1000

1500

2000

0.03 0.04 0.050

1000

2000

0.03 0.04 0.05

Figure 48: Off-momentum background in OPAL detector. Shown are distributions of theelectron energy normalized to the energy of the beam electrons (a,b), electron azimuthalangle (c,d), and electron polar angle (e,f), shown separately for the left (L) and right (R)side of the OPAL detector, for selected single-tagged events in 1999. The histograms arethe predictions for the single-tagged process from HERWIG, the off-momentum backgroundcontribution, and the background from other physics channels.

within about 10% for E > 0.7Eb. The genuine single-tagged events are complementedby the physics background and by artificially created single-tagged events constructed by acombination of anti-tagged two-photon events generated by PHOJET with a cluster createdby an off-momentum electron. The θ, φ and energy dependence of the electron clusters aregiven by the spectra of the additional clusters in Bhabha events.

Fig. 48 shows the 1999 OPAL data, which are expected to have the largest backgroundfrom off-momentum overlaps, compared with the prediction resulting from the sum of HER-WIG and the background, normalized to the luminosity of the data. Results are shown forθ, φ and energy of the scattered electron normalized to the energy of the beam electrons.Fig. 49 shows separately for the 1998, 1999 and 2000 OPAL data samples, the missing lon-gitudinal momentum and missing transverse momentum in the event, calculated includingthe untagged electron, which has been assumed to have zero transverse momentum and anenergy equal to the energy of the beam electrons. Both distributions have been normalized

83

∆pz/Ebev

ents

1998 a)

∆pT/Eb

even

ts

b)

OPALHERWIG+bgr.off-mom. bgr.physics bgr.

∆pz/Eb

even

ts1999 c)

∆pT/Eb

even

ts

d)

∆pz/Eb

even

ts

2000 e)

∆pT/Eb

even

tsf)

0

250

500

750

1000

-0.5 0 0.50

1000

2000

3000

0 0.02 0.04 0.06 0.08 0.1

0

500

1000

-0.5 0 0.50

2000

4000

0 0.02 0.04 0.06 0.08 0.1

0

500

1000

-0.5 0 0.50

1000

2000

3000

4000

0 0.02 0.04 0.06 0.08 0.1

Figure 49: Off-momentum background in OPAL detector. Shown are distributions of thescaled missing longitudinal momentum (a,c,e) and the scaled missing transverse momentum(b,d,f) in single-tagged events for the different years. The meaning of the histograms are asdefined in Fig. 48.

to the energy of the beam electrons. For energies below 0.6Eb the off-momentum backgroundclearly dominates and the observed angular dependences, especially in φ, clearly follow theexpected shape. The agreement between data and prediction is very good for all variablesexamined, providing confidence that the background from overlap of off-momentum beamelectrons is under control to a level of about 10%. The off-momentum background estimatewas used to calculate the contribution of fake double-tagged events, resulting from the over-lap of one background cluster with a single-tagged two-photon event and the overlap of twobackground clusters with an untagged event. In total 4.3, 15.2 and 4.6 overlap events arepredicted in OPAL data for the 1998, 1999 and 2000 data samples.

The total expected background from all the sources discussed above amounts to 42.2, 87,206.1, in case of OPAL, L3 and ALEPH, respectively.

84

9.3.2 Comparison with Monte Carlo models

PHOJET, TWOGAM, PYTHIA and PHOT02 Monte Carlo samples are used to correctthe data for acceptance and resolution effects in OPAL, L3 and ALEPH experiments. It istherefore essential that the shape of all important distributions is well reproduced by theseMonte Carlo simulations. Below a comparison is made of data distributions with predictionsfrom the above Monte Carlo models. Variables calculated from the scattered electrons as wellas variables calculated from the hadronic final state are studied. The integrated luminositiesof the Monte Carlo samples exceed many times that of the data.

OPAL: All Monte Carlo distributions shown are normalized to the data luminosity. InFig. 50 variables which are based on electron quantities, and the variables x1,2, are com-pared with predictions of PHOJET and background estimates. All variables, the normalizedelectron energies E1,2/Eb, the polar angles θ1,2, the azimuthal angles φ1,2, and photon virtu-alities Q2

1,2, are reasonably well described by the sum of the signal as predicted by PHOJETand the estimated background from overlaps with off-momentum electrons and other physicsprocesses. Note that PHOJET does not contain any explicit effects from BFKL, which wouldshow up in the region of low electron energies. Fig. 50e shows the logarithm of the ratioof the photon virtualities, log(Q2

1/Q22), of the two photons in an event. This distribution

is peaked around zero, indicating that the Q2 values of both photons are generally closeto one another, which is ideal to test for BFKL effects. In Fig. 51 distributions are shownwhich characterize the hadronic final state in double-tagged two-photon events: the numberof tracks, Nch, the visible hadronic invariant mass, Wvis, the hadronic energy, Ehad, the vari-able Y as well as the sum of the longitudinal

∑

pz and transverse∑

pT momenta. Withinstatistics, the agreement with PHOJET plus background estimates is reasonable.

L3: All Monte Carlo distributions shown are normalized to the number of data events.In Fig. 52 variables which are based on electron quantities, are compared with predictionsof PHOJET, TWOGAM and background estimates. All variables, the normalized electronenergies Ei/Eb, the polar angles θi, the photon virtualities Q2

i and their ratio ln(Q21/Q

22) are

reasonably well described by the sum of the signal as predicted by PHOJET or TWOGAMand the estimated background from other physics processes. However, TWOGAM givesslightly better description of the shape of θi and Q2

i distributions. The distributions ofthe visible hadronic invariant mass Wvis calculated using the four-vectors of all measuredparticles (see Eq. 9) and the corresponding variable Y vis are compared to Monte Carlopredictions on Fig. 53a and 53b. The same quantities but calculated using the scatteredelectrons four-momenta (see Eq. 11), Wee and Y ee, are shown in Fig. 53c and 53d. In allcases a good agreement with both Monte Carlo models is observed. It should be pointed outhere that due to undetected particles the distribution of Wvis (and in consequence Y vis) isshifted towards smaller values, whereas the distribution of Wee (Y ee) extends to much highervalues. However, as will be explained later, the distribution of Y ee is very sensitive to QEDradiative corrections, which are large.

ALEPH: All Monte Carlo predictions shown are normalized to the data, that means thecross-section predicted by PYTHIA was reduced by 12% and the cross section predicted by

85

E1,2/Eb

even

ts

OPALPHOJET+bgr.off-mom. bgr.physics bgr.

a)

Q2 1,2 [GeV2]

even

ts

b)

θ1,2 [rad]

even

tsc)

φ1,2 [rad]

even

ts

d)

log(Q12/Q2

2)

even

ts

e)

x1,2

even

ts

f)

0

20

40

60

80

0.4 0.6 0.8 10

50

100

150

0 10 20 30 40

0

50

100

0.03 0.04 0.05 0.060

20

40

60

80

0 2 4 6

0

20

40

60

-1 -0.5 0 0.5 10

25

50

75

100

0 0.1 0.2 0.3 0.4 0.5

Figure 50: Comparison of OPAL data with Monte Carlo predictions. Shown are distributionsof (a) the energies of the electrons, normalized to the beam energy, (b) the virtualities ofboth photons, (c) the polar angles of the electrons, (d) the azimuthal angles of the electrons,(e) the ratio of the photon virtualities and (f) the x values of double-tagged two-photonevents. The histograms are the predictions for the double-tagged two-photon process fromPHOJET1.10, the off-momentum background contribution, and the background from otherphysics channels.

PHOT02 was increased by 30%. In Fig. 54 we show the distributions of the normalizedelectron energies E1,2, the polar angles θ1,2, the azimuthal angle between the electron scat-tering planes ∆Φ, the photon virtualities Q2

1,2 and their ratio, the number of tracks Nch, the

hadronic invariant mass (denoted as Wγγ) and the variable Y . Both Monte Carlo models,PYTHIA and PHOT02, describe the shape of the above variables in the data equally well.Only in case of the quantity ∆Φ, which in the data is flat, the prediction of PYTHIA issignificantly better.

From the above Monte Carlo models used by the experiments to correct the data fordetector effects only TWOGAM contains QED radiative corrections to the lepton lines.Therefore it is necessary to estimate independently the size of radiative corrections in doubletagged events. We used TWOGAM and BDK [73] Monte Carlo generators which agree in

86

OPALPHOJET+bgr.off-mom. bgr.physics bgr.

Nch

even

ts

a)

Wvis [GeV]

even

ts

b)

Ehad [GeV]

even

tsc)

Y––

even

ts

d)

Σpz [GeV]

even

ts

e)

ΣpT [GeV]

even

ts

f)

0

20

40

60

0 10 2010

-210

-11

10

10 2

0 20 40 60 80

0

20

40

60

0 10 20 30 400

20

40

60

80

0 2 4 6

0

20

40

60

-40 -20 0 20 400

20

40

60

0 2 4 6

Figure 51: Comparison of OPAL data with Monte Carlo predictions. Shown are distributionsof (a) the number of tracks of the hadronic final state, (b) the visible hadronic invariant mass,(c) the total hadronic energy (d) the variable Y , (e) the missing longitudinal momentumand (f) the missing transverse momentum for double-tagged events. The PHOJET1.10predictions and backgrounds contributions are as in Fig. 50.

description of radiative corrections for the process e+e− → e+e− µ+µ−. The size of theradiative corrections depends on the variables used to calculate the kinematics, and also tosome extent on the non-radiative cross-section. As an example we present the estimation ofradiative corrections by the OPAL experiment. This study has been performed purely onthe generator levels, without detector simulation. In radiative events the scattered electronsare combined with the closeby photons that would also not be resolved in the experimentalsetup (due to granularity of the calorimeters). The combined object is considered to be theobserved electron and is used to calculate the kinematic variables. Hence the data are mainlysensitive to initial state radiation. The radiative corrections have been estimated in bins ofY , as the ratio of the relative difference of the non-radiative and radiative cross sections withrespect to the radiative cross section, while applying the experimental restrictions on thescattered electrons energies and polar angles as well as on the minimum invariant mass ofthe hadronic system. Figure 55 compares the effect of radiative corrections in two methodsof calculating the variable Y . The first method uses the hadronic final state to calculate W

87

0

100

200

300

0.4 0.6 0.8 10

50

100

150

10 20 30 40

0

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200

0.04 0.06

Ei/Eb

Ent

ries/

0.02

5

a)

Q2 i [GeV2]

Ent

ries/

2 G

eV2

b) L3 dataPHOJETTWOGAMBack.

θi [rad]

Ent

ries/

4 m

rad

c)

log(Q2 1 /Q

2 2)

Eve

nts/

0.2

d)

0

50

100

150

-1 -0.5 0 0.5 1

10

10 2

10 3

0 10 20 30 0

50

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-2 0 2 4 6

10

10 2

0 25 50 75 100

W vis [ GeV ]

Eve

nts/

2.5

GeV

a) L3 data PHOJET TWOGAM Back.

Y vis

Eve

nts/

0.5

b)

W ee [ GeV ]

Eve

nts/

10 G

eV

c)

Y ee

Eve

nts/

0.5

d)

0

25

50

75

100

0 2 4 6 8

Figure 52: Comparison of L3 data with Monte Carlo predictions. Distributions of (a) theenergies of the electrons, normalized to the beam energy, (b) the virtualities of both photons,(c) the polar angles of the electrons, (d) the ratio of the photon virtualities for double-taggedevents. The histograms are the predictions for the double-tagged two-photon process fromPHOJET and TWOGAM and background from other physics channels.

Figure 53: Comparison of L3 data with Monte Carlo predictions. Distributions of themeasured two photon invariant mass W and the corresponding variable Y , calculated fromthe detected particles in the hadronic final state (a,b) and from the scattered electrons (c,d).The PHOJET and TWOGAM predictions and background contributions are as in Fig. 52.

(Eq. 9). The second method uses the beam and scattered electrons to obtain W (Eq. 11).The predictions on the size of radiative corrections from different subprocesses generatedby TWOGAM are shown separately. Also the predictions weighted with the relative crosssections of different subprocesses shown in Fig. 55c are shown in the figure. For the fullyhadronic method the radiative corrections are small. However, for the electron methodthe corrections can be larger than 50% at large Y values. Obviously, measurements basedon the electron kinematics cannot be compared with models or BFKL calculations in theregion Y > 4, unless radiative corrections have been applied. Since the actual size of theradiative corrections also depends on the non-radiative cross-section itself, in principle aniterative procedure would be required to extract the non-radiative cross-section. However,the present statistics of data do not permit such a procedure. In the analyses presented inthis paragraph only the L3 experiment used the electron method to calculate W (and inconsequence Y ) and corrected the data for radiative effects. OPAL and ALEPH used thefully hadronic method to calculate Y where the effects of radiative corrections are small.

88

0

2

4

6

8

10

0.4 0.6 0.8 1

ALEPH DATAPYTHIAPHOT02γγ → ττ

–

off momentumqq

–

E’1,2 / Ebeam

dσvi

s/d(E

’ 1,2

/ Ebe

am)

[pb]

0

20

40

60

80

100

0.05 0.075 0.1 0.125 0.15

ALEPH DATAPYTHIAPHOT02γγ → ττ

–

off momentumqq

–

θ1,2 [deg]

dσvi

s/dθ 1,

2 [p

b/de

g]

0

0.2

0.4

0.6

0.8

1

0 1 2 3

ALEPH DATAPYTHIAPHOT02γγ → ττ

–

off momentumqq

–

∆Φ [rad]

dσvi

s/d∆Φ

[pb/

rad]

10-3

10-2

10-1

50 100 150 200

ALEPH DATAPYTHIAPHOT02γγ → ττ

–

off momentumqq

–

Q1,22 [GeV2]

dσvi

s/dQ

1,22 [p

b/G

eV2 ]

0

0.1

0.2

0.3

0.4

0.5

0.6

-3 -2 -1 0 1 2 3

ALEPH DATAPYTHIAPHOT02γγ → ττ

–

off momentumqq

–

-log(Q12/Q2

2)

dσvi

s/d(-

log(

Q12 /Q

22 )) [p

b]

0

0.05

0.1

0.15

0.2

0.25

0.3

5 10 15 20

ALEPH DATAPYTHIAPHOT02γγ → ττ

–

off momentumqq

–

Nch

dσvi

s/dN

ch [p

b]

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6

ALEPH DATAPYTHIAPHOT02γγ → ττ

–

off momentumqq

–

pt [GeV]

dσvi

s/dp t [

pb/G

eV]

10-3

10-2

10-1

20 40 60 80

ALEPH DATAPYTHIAPHOT02γγ → ττ

–

off momentumqq

–

Wγγ [GeV]

dσvi

s/dW

γγ [p

b/G

eV]

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8

ALEPH DATAPYTHIAPHOT02γγ → ττ

–

off momentumqq

–

Y=ln(Wγγ2 / (Q1

2Q22)1/2)

dσvi

s/dY

[pb]

Figure 54: Comparison of ALEPH data with Monte Carlo predictions. Distributions of (a)the energies of the electrons, normalized to the beam energy, (b) the polar angles of theelectrons, (c) the azimuthal angle between the electron scattering planes ∆Φ, (d) the virtu-alities of both photons, (e) the ratio of the photon virtualities, (f) the number of tracks, (g)total transverse momentum in the event, (h) the hadronic invariant mass and (i) the variableY , are shown for double tagged events. The histograms are the predictions for the double-tagged two-photon process from PYTHIA and PHOT02, the off-momentum backgroundcontribution, and the background from other physics channels.

9.3.3 Results

Several differential e+e− and total γ⋆γ⋆ cross sections have been measured by the experi-ments. Due to different geometrical acceptances of the detectors, the measurements havebeen performed in slightly different phase space regions presented in Table 12 (in order not tobias the measurements by model predictions the phase space in which the cross sections havebeen measured is close to the kinematical selection cuts used at the detector level). Belowthe main results obtained by OPAL, L3 and ALEPH will be presented in turn. Due tolimited statistics in the data, a simple bin-by-bin method was applied to correct for detectorand selection inefficiencies. The efficiency, Re, and purity, Rp, are defined as:

Re =NDet⊗Had

NHadRp =

NDet⊗Had

NDet

where NDet⊗Had is the number of events which are generated in a bin and measured in thesame bin, NHad is the number of events which are generated in a bin and NDet is the number

89

Y––

(σbo

rn-σ

rad)

/σra

d

full qpm qcd vmd a)

Y––

(σbo

rn-σ

rad)

/σra

db)

Y––

rel.

frac

. [%

]

c)

-50

0

50

0 1 2 3 4 5 6 7

-50

0

50

0 1 2 3 4 5 6 7

0

0.25

0.5

0.75

1

0 1 2 3 4 5 6 7

Figure 55: Radiative corrections for the process e+e− → e+e− hadrons as a function of Y fortwo different methods to calculate W : a hadronic method (a) and an electron method (b), asexplained in the text. Shown are the relative differences of the non-radiative and radiativecross sections with respect to the radiative cross section as a function of Y for differentsubprocesses: QPM (open circle), QCD (open square) and VMD (open triangle) as well asfor their weighted average (full circle). In (c) the relative fraction of different subprocesses,as predicted by TWOGAM, in the total cross section are shown as a function of Y .

of events measured in a bin. In both definitions the terms ‘generated’ and ‘measured’ denoteevents which pass all selection cuts at the hadron or at the detector level, respectively. Thecorrection factor NHad/NDet is obtained by dividing purity by efficiency. For the W variablethe purity is typically around 60% over the whole range, and the efficiency is in the rangeof 30-50%. Similar numbers are obtained for Y and x, while for the ∆φ and Q2 variablesthe efficiencies are around 60% and purities around 80%. The correction factor is typicallyaround 1.5 and fairly constant. From the measurement of the differential cross-sections of theprocess e+e− → e+e− hadrons we extract the cross-sections for the reaction γ⋆γ⋆ → hadronsas a function of the variable under study, using LTT (see Eq. 72), obtained separately foreach bin using Monte Carlo. Technically this was done by setting σγ⋆γ⋆ ≡ 1 in GALUGAand integrating (Eq. 71) using only LTT for each bin within the experimental phase spacerestrictions. The cross-sections for the reaction γ⋆γ⋆ → hadrons predicted by the models arecalculated using the same LTT factors.

90

Phase space for the cross section measurements

OPAL ALEPH L3

Ei > 0.4Eb Ei > 0.3Eb Ei > 40 GeV34 < θi < 55 mrad 35 < θi < 155 mrad 30 < θi < 66 mradW > 5 GeV W > 3 GeV W > 5 GeV

Table 12: Phase space used for the cross section measurements for the process e+e− →e+e− hadrons, by different LEP experiments.

The QPM cross-section for the process e+e− → e+e− qq, which corresponds to the dia-gram labelled F0 in Fig. 20, was calculated with the GALUGA [65] program, which includesall terms from Eq. 13. The quark masses assumed are 0.325 GeV for uds and 1.5 GeV forc quarks. For the region of W > 5 GeV considered here, the cross-section depends onlyweakly on the chosen masses, e.g. the results for u and c quarks differ only slightly.

OPAL: Differential e+e− and total γ⋆γ⋆ cross-sections are obtained as functions of x, Q2,W , and the azimuthal correlation between the two scattered electrons ∆φ. Here Q2 refers tothe maximum of Q2

1 and Q22, and x is the corresponding value of x1 or x2. For the comparison

with BFKL predictions we also present the differential cross-section as a function of Y . Forall variables except Q2 the results for the cross-sections of the reaction γ⋆γ⋆ → hadronsare given at an average value of Q2, 〈Q2〉 = 17.9 GeV2. The main contributions to thesystematic errors come from changing the Monte Carlo model from PHOJET to PYTHIA(−16% change in the total cross-section), changing the cut on Nch (−12%), and varyingthe lower cuts on W (±8%) and θ1,2 (±5%). The normalization uncertainty due to theluminosity measurement is less than 1% and has been neglected.

Radiative corrections are calculated with the program BDK [66]. As for the GALUGAMonte Carlo, the BDK program calculates the QPM cross-section with, in addition, initialand final state QED radiative corrections to the scattered electrons. It has been verifiedthat the non-radiative cross-sections predicted by BDK and GALUGA agree with each other.GALUGA has more flexibility for calculating cross-sections and is therefore used to calculatethe LTT factors and QPM predictions. The results are presented in Fig. 56, 57 and 58 andin Table 32. In Fig. 56 we show the measured differential cross-section for the processese+e− → e+e− hadrons and the cross-section for γ⋆γ⋆ → hadrons as a function of x andQ2. PHOJET better describes the data at lower x values, where the QPM prediction is toolow. In contrast, in the large x region the QPM prediction is sufficient to account for thedata. Fig. 57 shows the measured cross-sections as a function of W and ∆φ. The modelpredictions indicate a slightly different shape than is observed for the data. Studies forHERA have shown [76] that angular variables similar to ∆φ can be sensitive to the presenceof BFKL dynamics, but so far no calculations are available for γ⋆γ⋆ scattering. The datashow that the ∆φ behavior of the cross-section for the reaction e+e− → e+e− hadrons isflat, while the cross-section for the process γ⋆γ⋆ → hadrons increases from ∆φ = 0 to∆φ = π. PHOJET1.10 does not describe the ∆φ distribution, whereas QPM reproducesthe shape of the distribution. One should remark that the earlier version, PHOJET1.05,follows the data in both shape and normalization. In Fig. 58 we compare the measuredcross-section for the processes e+e− → e+e− hadrons and γ⋆γ⋆ → hadrons as a function of Y

91

with the PHOJET Monte Carlo, the QPM calculation, the NLO calculation for the reactione+e− → e+e− qq, and numerical BFKL calculations. The BFKL predictions are shown forthe LO-BFKL [41] and a NLO-BFKL [42] calculation both using Y . Also shown is a (partial)HO-BFKL calculation [43] using Y . All BFKL predictions are shown for Y or Y > 2, exceptfor the NLO-BFKL calculation which has been evaluated for Y > 1. Both PHOJET andQPM using massive quarks describe the data equally well. Also the NLO calculation for thereaction e+e− → e+e− qq, evaluated for five massless quarks and using ΛMS

5 = 0.2275 GeV,is in accord with the data. As can be seen from Table 13 the predicted differential cross-sections as functions of Y and Y are very different at small values, but get much closer athigher values. This is expected from the approximation made in Eq. 81. It means thatat low values of Y the comparison of the experimental result with predictions based on Yis rather uncertain, whereas at high values the uncertainty from using different definitionsis small. For all BFKL predictions shown the cross-section is significantly larger than the

Y or Y dσ/dY dσ/dYrange LO NLO LO NLO0 – 1 0.071 0.065 0.015 0.0141 – 2 0.135 0.128 0.149 0.1382 – 3 0.087 0.089 0.111 0.1223 – 4 0.041 0.047 0.047 0.0534 – 6 0.010 0.013 0.011 0.014

Table 13: Predictions for the differential cross-section for the process e+e− → e+e− qq asfunctions of Y and Y in LO and NLO using the calculation from [34].

PHOJET prediction for Y > 3, and the differences increase with increasing Y . The LO-BFKL calculation predicts a cross-section which is too large compared to the data. ThisLO-BFKL calculation (Bartels99) [41] already incorporates improvements compared to theoriginal results [45] by including effects of the charm quark mass, the running of the strongcoupling constant αs and contribution of longitudinal photon polarization states. HenceBFKL effects as large as predicted by the LO-BFKL calculation are not in agreement withthe data. BFKL cross-sections have been calculated to NLO (Kim99) [42], using the BLM [77]optimal scale setting. At the highest Y value the NLO-BFKL cross-section is a factor sevenlarger than the PHOJET prediction. The data lie in between these two predictions. Finally,the calculation (Kwiecinski) [43] contains the dominant contribution of the higher ordercorrections via the so-called consistency constraint, to all orders. Its prediction in the highestreachable Y range is about a factor two lower than for the NLO-BFKL calculation, and thisprediction and PHOJET are roughly equally compatible with the data.

The total measured cross-section for the process e+e− → e+e− hadrons in phase spacedefined in Table 12, is 0.35 ± 0.04 (stat) +0.04

−0.08 (sys) pb. The expected cross-section fromPHOJET is 0.39 ± 0.02(stat) pb, while the GALUGA prediction for QPM using massivequarks is 0.27 ± 0.02(stat) pb, and the NLO predictions for the reaction e+e− → e+e− qqusing massless quarks is 0.35 pb.

L3: Differential e+e− and total γ⋆γ⋆ cross sections are obtained as functions of Q2 =√

Q21 Q2

2,W and Y . They are presented in Fig. 59 and Fig. 60 and listed in Table 35. The ranges

92

x

dσ/d

x [p

b]

a)

OPALQPMPHOJET 1.10

x

σ γ✶γ✶

(x)

[nb]

b)

Q2 [GeV2]

dσ/d

Q2 [p

b/G

eV2 ]

c)

Q2 [GeV2]

σ γ✶γ✶

(Q2 )

[nb]

d)

0

0.5

1

1.5

2

0 0.1 0.2 0.30

5

10

15

0 0.1 0.2 0.3

0

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0.02

0.03

0.04

10 15 20 250

5

10

15

10 15 20 25

W [GeV]

dσ/d

W [p

b/G

eV]

OPAL

QPM

PHOJET 1.10

a)

W [GeV]

σ γ✶γ✶

(W)

[nb]

b)

∆φ [rad]

dσ/d

∆φ [p

b/ra

d]

c)

∆φ [rad]

σ γ✶γ✶

(∆φ)

[nb]

d)

10-4

10-3

10-2

0 20 40

2

4

6

8

10

12

0 20 40

0

0.05

0.1

0.15

0.2

0.25

0 1 2 30

5

10

15

0 1 2 3

Figure 56: Cross-sections for the process e+e− → e+e− hadrons in the phase space regiondefined in Table 12, and for the process γ⋆γ⋆ → hadrons, as functions of x for 〈Q2〉 = 17.9GeV2 (a,b), and as functions of Q2 (c,d). Data are shown as full dots in the centre of thebins. The inner error bars represent the statistical errors and the outer error bars representstatistical and systematic errors added in quadrature. The predictions of PHOJET1.10 areshown as solid lines, and those of QPM as dashed lines.

Figure 57: Cross-sections for the process e+e− → e+e− hadrons in the phase space regiondefined in Table 12, and for the process γ⋆γ⋆ → hadrons for 〈Q2〉 = 17.9 GeV2, as functionsof W (a,b) and ∆φ (c,d). The meaning of symbols as in Fig. 56.

10 GeV2 < Q2 < 32 GeV2, 5 GeV < W < 100 GeV and 2 < Y < 7 are investigated in-dependently. An important difference between the L3 analysis and the analyses by OPALand ALEPH is the calculation of W , which in case of L3 was obtained from the scatteredelectrons via Eq. 11. In consequence, in the L3 analysis the QED radiative corrections werelarge and were estimated by TWOGAM using the method described above. The systematicuncertainty on the cross sections due to the selection cuts is 5%. The uncertainty due tobackground estimation of single tagged events is 3.4%. The uncertainty due to Monte Carlomodeling is estimated as 6.4% by comparing PHOJET and TWOGAM with QED radiativecorrections switched off. The uncertainty in the estimation of radiative corrections was deter-mined as 3% by comparing the size of radiative corrections to the process e+e− → e+e− µ+µ−

predicted by TWOGAM and RADCOR programs. The different contribution from QPM,VMD and QCD subprocesses as function od Y and W gives an additional systematic un-certainty. A 20% variation of the QCD component results in an uncertainty of 0.3% at lowvalues and of 5.7% at large values of Y and W . This uncertainty is about 0.5% over thefull Q2 range. The cross section for the process e+e− → e+e− hadrons after applying QEDradiative corrections are compared in Fig59 to the PHOJET Monte Carlo and to LO andNLO calculations of γ⋆γ⋆ → qq [34]. In these calculations the mass of quarks is set to zeroand α is fixed to the value for on-shell photons. The calculations describe well the Q2 depen-

93

Y––

dσ/d

Y–– [p

b]

a) OPAL

PHOJET 1.10QPMKwiecinski et al.Cacciari et al. (NLO)

Y––

σ γ✶γ✶

(Y––)

[nb]

b)Kim99Bartels99

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 1 2 3 4 5 6

02468

101214161820

0 1 2 3 4 5 6

Figure 58: Cross-sections for the process e+e− → e+e− hadrons in the region E1,2 > 0.4Eb,34 < θ1,2 < 55 mrad and W > 5 GeV, and the process γ⋆γ⋆ → hadrons for 〈Q2〉 = 17.9GeV2, as functions of Y . Data are shown as full dots in the centre of the bins. The innererror bars represent the statistical errors and the outer error bars represent statistical andsystematic errors added in quadrature. The predictions of PHOJET1.10 are shown as thesolid lines, that of the NLO calculation of the process e+e− → e+e− qq as dotted lines,and those of QPM as dashed lines. Three BFKL calculations are shown: a LO one fromBartels et al. (Bartels99), NLO from Kim et al. (Kim99) using Y , and the calculation fromKwiecinski et al., using the consistency constraint calculated for Y .

dence observed in data. For the W and Y distributions, the QPM calculations describe thedata except in the last bin, where the measured cross section exceeds the predictions. Suchan excess is expected if the resolved photon QCD processes become important at large Y .The predictions of PHOJET, which includes the QPM and QCD processes in the frameworkof DGLAP equations, also describe the data. A similar behaviour may also be obtainedby considering ‘hard pomeron’ contribution in the framework of BFKL theories, while LOBFKL calculations we found to exceed the experimental values by a large factor.

A fit of the form f(Q2) = A/Q2 predicted by perturbative QCD [45,46] to the measuredγ⋆γ⋆ cross section as a function of Q2 gives A = 81.8 ± 6.4 nb/GeV2 and χ2/d.o.f. = 1.2/3.The average value of σγ⋆γ⋆ in the kinematical region given in Table 12 is 4.7±0.4 nb.The NLOcalculations [34] predict a decrease of σγ⋆γ⋆ as a function of W or Y , which is inconsistentwith the measurements at large values of W and Y .

94

10-2

10-1

10 15 20 25 30

10-4

10-2

1

20 40 60

L3

Q2 [GeV2]

dσee

/dQ

2 [pb/

GeV

2 ]

a) DataQPMNLO γ∗ γ∗ →qq

––

PHOJET

L3

b)

Wγγ [GeV]

dσee

/dW

γγ [p

b/G

eV]

L3

c)

Y=ln(W2γγ /Q

2)

dσee

/dY

[pb]

10-3

10-2

10-1

1

3 4 5 6

0

5

10

15 20 25 30

0

5

10

20 40 60

Data1/Q2 FITNLO γ∗ γ∗ →qq

––

a)

L3

Q2 [GeV2]

σ γ∗γ∗

[nb]

Wγγ [GeV]

σ γ∗γ∗

[nb]

b) DataNLO γ∗ γ∗ →qq

––

L3

DataNLO γ∗ γ∗ →qq

––c)

L3

Y=ln(W2γγ /Q

2)

σ γ∗γ∗

[nb]

0

5

10

2 3 4 5 6

Figure 59: The differential cross section for the process e+e− → e+e− hadrons in the kine-matical region defined in the text, measured by the L3 experiment. Points represent the datawith statistical and systematic errors added in quadrature. The LO and NLO predictionsof [34] for the process γ⋆γ⋆ → qq are displayed as the dashed and solid lines, respectively.The dotted line shows the prediction of the PHOJET Monte Carlo.

Figure 60: Cross section for the process γ⋆γ⋆ → hadrons in the kinematical region definedin the text, measured by the L3 experiment. The dashed line represents the fit to the datadescribed in the text. The NLO predictions of [34] for the process γ⋆γ⋆ → qq are displayedas the solid line.

ALEPH: Differential e+e− cross sections are obtained as functions of the energy of thescattered electrons scaled by the energy of beam electrons E1,2/Eb, the polar angles θ1,2, theazimuthal angle between the two electron scattering planes ∆Φ, photon virtualities Q2

1,2, two

photon invariant mass Wγγ and the variable Y . They are presented in Fig. 61 and listed inTable 33 and 34. In order to estimate the systematic effects caused by an imperfect detectorsimulation all subdetector resolutions were scaled by 10%. The uncertainty due to a possibleshift in the energy scale of the small angle calorimeters was estimated by introducing anoffset of 0.5 GeV to the measure energy. the polar and azimuthal angles of the scatteredelectrons were shifted by 0.25 mrad and 0.5 mrad, respectively. The cross sections of thebackground processes were changed conservatively by ±10%. The dependence on the MonteCarlo model used to correct the data was estimated using PHOT02 instead of PYTHIA. Thesystematic error was calculated as the quadratic sum of the three largest effects in each bin.The main contributions to the systematic error came from varying the resolutions for the

95

0

10

20

30

40

0.4 0.6 0.8 1

ALEPH DATAPYTHIAPHOT02NLO QCD

E’1,2 / Ebeamdσ

/d(E

’ 1,2

/ Ebe

am)

[pb]

10

10 2

0.05 0.075 0.1 0.125 0.15

ALEPH DATAPYTHIAPHOT02NLO QCD

θ1,2 [rad]

dσ/d

θ 1,2

[pb/

rad]

0

0.5

1

1.5

0 1 2 3

ALEPH DATAPYTHIAPHOT02NLO QCD

∆Φ [rad]

dσ/d

∆Φ [p

b/ra

d]

10-3

10-2

10-1

50 100 150 200

ALEPH DATAPYTHIAPHOT02NLO QCD

Q2 1,2 [GeV2]

dσ/d

Q2

1,

2 [p

b/G

eV2 ]

10-3

10-2

10-1

20 40 60 80

ALEPH DATAPYTHIAPHOT02NLO QCD

Wγγ [GeV]

dσ/d

Wγγ

[pb/

GeV

]

0

0.2

0.4

0.6

0.8

0 2 4 6

ALEPH DATAPYTHIAPHOT02NLO QCD

Y=log(W2γγ / (Q1

2Q22)1/2)

dσ/d

Y [p

b]

Figure 61: The differential cross section for the process e+e− → e+e− hadrons in the kinemat-ical region defined in Table 12, measured by the ALEPH experiment. Points represent thedata with statistical and systematic errors added in quadrature. The predictions of PYTHIA(dash), PHOT02 (dash-dot) and NLO predictions of [34] for the process e+e− → e+e− qq(solid) are shown as histograms.

hadronic system and the tagged electron energy. The measured cross sections are comparedwith the PYTHIA and PHOT02 Monte Carlo models and with a NLO calculation of [34]for the process e+e− → e+e− qq. Due to the geometrical acceptance there is a gap in themeasurement of the cross section as a function of θ1,2 between 55 mrad and 60 mrad. How-ever, for the other measurements this gap has been interpolated with a Monte Carlo model.The presented differential cross sections are rather well described by the two Monte Carlomodels and by the NLO calculation. The only exception is the azimuthal angle betweenelectron scattering planes. In this case only PYTHIA gives a good description of the data.The NLO calculation underestimates the cross section at large angles, whereas PHOT02does not describe the data.

Due to the large range of the possible polar angles of the scattered electrons in theALEPH measurements, the virtualities of exchanged photons can be quite different, allow-ing for an evolution in transverse momenta of gluons emitted along the ladder in Fig. 19.

96

0

0.2

0.4

0.6

0.8

1

-3 -2 -1 0 1 2 3

ALEPH DATAPYTHIAPHOT02NLO QCD

-log(Q12/Q2

2)

dσ/d

(-lo

g(Q

12 /Q22 ))

[pb]

0

5

10

15

20

25

0 1 2 3 4 5 6Y=log(Wγγ

2 / (Q12Q2

2)1/2)

σ γγ (

Y)

[nb] ALEPH DATA

BFKL LO

BFKL NLO

Figure 62: The differential cross section for the process e+e− → e+e− hadrons in the kine-matical region defined in Table 12, measured by the ALEPH experiment, as a function oflog Q2

1/Q22. The meaning of symbols as in Fig. 61.

Figure 63: The total cross section for the process γ⋆γ⋆ → hadrons measured by the ALEPHexperiment, as a function of Y . The data are compared with LO BFKL and NLO BFKLpredictions, with the bands giving the uncertainty from the variation of the scale parameters0 from Q2 to 10 Q2 in LO and from Q2 to 4 Q2 in NLO.

This can be seen in Fig. 62 where the differential e+e− cross section as the function of theratio of the virtualities of the exchanged photons in the phase space defined in Table 12 ispresented. Therefore the above ALEPH results can not be compared with BFKL calcula-tion. To enable this comparison a further cut has been applied, | log Q2

1/Q22| < 1. The whole

analysis, including the systematic error estimation, was redone with this additional cut. Theagreement between the ALEPH data and the Monte Carlo simulations was on the same levelas previously, while the data statistics was reduced by 40%. In Fig. 63 we present the γ⋆γ⋆

cross section as a function of Y in the phase space defined with cuts in Table 12 togetherwith the additional cut on the photon virtualities, | log Q2

1/Q22| < 1. The measured cross

section is compared to LO and NLO BFKL calculations. The ranges denoting the LO andNLO calculations correspond to the variation of the scale parameter s0 from Q2 to 10 Q2 orfrom Q2 to 4 Q2, respectively. The LO calculations overestimate the measured cross sectioneven at lower values of Y . The NLO calculation is consistent with the data (although somesystematic shift is observed).

After the presentation of independent results obtained by different LEP experiments onthe structure of interactions of virtual photons, it is time to ask whether they are consistent.It is not easy to compare them because the phase space of each measurement (see Table12) is different, and also the observables for which the cross sections have been measuredare not the same. Essentially the only quantity which can be compared is the differentiale+e− cross section as a function of Y . However, due to different phase space regions of themeasurements it is necessary to use Monte Carlo models to ‘correct’ the measurements tothe same phase space. We have chosen the phase space of the OPAL measurement and usedPYTHIA (for ALEPH) and PHOJET (for L3) Monte Carlo programs to find the correctionfactors. The comparison is shown in Fig. 64. The original errors of the L3 and ALEPH

97

Y––

dσ/d

Y–– [p

b]

OPAL

L3

ALEPH

PHOJET 1.10

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 1 2 3 4 5 6 7

Figure 64: Comparison of the cross sections for the process e+e− → e+e− hadrons as afunction of Y measured by different LEP experiments. The ALEPH and L3 data have beencorrected to the phase space used in the OPAL measurement using Monte Carlo models asexplained in the text.

measurements have been increased by the (small) statistical errors of the correction factors.One can conclude that the agreement between the LEP measurements is quite good. Thedata are consistent with PHOJET predictions which does not involve any BFKL like effectsand with existing NLO or HO BFKL predictions. However, for unambigous confirmation ofBFKL effects in interactions of two virtual photons one needs to acces higher values of Y ,which migh be possible only at a future linear collider.

98

10 Summary and outlook

In this review article we discussed in detail the theoretical foundations and the measurementsof the QED and hadronic structure of the real and virtual photons as well as the hadronicstructure of the electron. The paper is concentrated on the measurements performed atLEP experiments, although wherever similar measurements performed at other e+e− collidersexist, they are compared to the LEP results. The measurements at LEP have been performedat several e+e− centre-of-mass energies starting from

√see ≈ 92 GeV at LEP1 phase (with

integrated luminosities of about 100 pb−1 per experiment), and in the range√

see = 161−209GeV at LEP2 phase (with integrated luminosities of above 600 pb−1 per experiment).

The measurements of the QED structure of the photon are based on full LEP1 statistics.Extending the measurements to LEP2 phase would reduce the errors by a factor of more thantwo, however this is not planed. Together with the measurements performed by the otherexperiments, the QED structure function of the quasi-real photon F γ

2,QED has been measuredin the range of average probe photon virtualities of 0.43 < 〈Q2〉 < 130 GeV2. The effect ofthe small, but different from zero, virtuality of the target photon has been observed. Apartof F γ

2,QED, two other structure functions of the real photon, F γA,QED and F γ

B,QED, have beenmeasured at LEP for the first time. The QED structure of interactions of virtual photonshas been also studied at LEP, and the importance of the contribution to the cross sectionof the interference terms, τTT and τTL, was observed. All these measurements are in a verygood agreement with predictions of QED.

The hadronic structure function of the photon F γ2 has been measured at LEP using

the full available statistics. Together with several measurements performed at other e+e−

colliders, the data cover at present a broad range of average virtualities, 0.24 < 〈Q2〉 < 780GeV2 and reach as low x as approximatelly 10−3. The data clearly disfavour the QPMpredictions, and suggest a rise of F γ

2 towards low values of x, as predicted by most of themodern parameterisations of F γ

2 . Using the broad range of 〈Q2〉, the Q2 evolution of F γ2 was

studied. The expected positive scaling violations for all values of x have been confirmed.The hadronic structure function of the photon has been also measured separately for charmquarks using full LEP2 statistics. The measured F γ

2,c agrees perfectly with perturbative QCDcalculation at NLO for x > 0.1, and is above the predictions for x < 0.1.

As suggested by some theorists, the hadronic structure function of the electron F e2 has

been recently measured at LEP. Because of the much better reconstruction of the variable zthan the variable x, the measurement of F e

2 does not require an unfolding procedure, which isneeded for the measurement of F γ

2 . However, because the different predictions of the photonstructure function lead to very similar electron structure functions, the measurement of F e

2

does not have a better resolving power than the measurements of F γ2 .

The hadronic structure of interactions of virtual photons have been studied in terms ofthe effective structure function of the virtual photon F γ

eff and differential e+e− and total γ⋆γ⋆

cross sections. The measurements of F γeff are in full agreement with the naive QPM model

predictions derived from the doubly virtual box diagram. The measured P 2 dependenceof F γ

eff, as expected, suggests a slow decrease with increasing P 2. The measurements ofthe cross sections for interactions of two highly virtual photons serve as an ideal tool todistinguish between DGLAP and BFKL dynamics of partons inside the photon, because the

99

1

10

10 102

103

104

F2 γ

x5.6 10-4

(× 4.0)

1.8 10-2

(× 2.8)

5.6 10-3

(× 2.0)

1.8 10-2

(× 1.5)

5.6 10-2

LEP 2

LC 500θ0 = 175 mr

Q2(GeV2)

x0.8

(× 4.0)

0.6(× 2.8)

0.4(× 2.0)

0.25(× 1.5)

0.14

10 102

103

104

1

10

10 102

103

104

F2 γ

x5.6 10-4

(× 4.0)

1.8 10-3

(× 2.8)

5.6 10-3

(× 2.0)

1.8 10-2

(× 1.5)

5.6 10-2

Q2(GeV2)

x0.8

(× 4.0)

0.6(× 2.8)

0.4(× 2.0)

0.25(× 1.5)

0.14

LC 500θ0 = 40 mr

10 102

103

104

Figure 65: Simulated linear collider data for integrated luminosity 10 fb−1 at√

see = 500GeV if tagging is feasible for θ > 175 mrad (upper plots) and for θ > 40 mrad (bottomplots). The LEP2 points shown in the upper plots are simulated data points.

appropriate calculations do not involve any non-perturbative input. The measured crosssection σγ⋆γ⋆ as a function of Y disfavours as steep rise of the cross section as predicted bythe LO BFKL calculation. The data are consistent with NLO BFKL calculation. However,the NLO corrections are very large, and therefore higher order calculations are needed tomake the perturbative BFKL calculations more credible.

The measurements of the photon structure functions are expected to be extended tohigher values of Q2 and to lower values of x at a future linear collider. The discussion ofthe physics goals for a linear collider can be found e.g. in [154,155]. A future linear colliderrunning in e+e− mode is expected to operate in the first phase at

√see = 500 GeV and to

100

Polarized e-beam

Spot size for soft γ

Spot size for hard γSpent electrons deflectedin a magnetic field

Polarized laser beam

Figure 66: Creation of high energetic photons in the Compton backscattering process oflaser photons off the beam electrons. The figure is taken from [158].

be able to accumulate an integrated luminosity of 100 fb−1 per year of operation. In thesecond phase the energy is expected to be rised to

√see = 2 TeV. The background studies

performed in [157] show that the amount of background per bunch crossing expected for theTESLA design is a factor of 106 higher than the number of two photon interactions withW > 5 GeV and with an average visible energy of 10 GeV. Most of background sources willlead to e+e− pair creation and to hadronic background, with most of particles produced inthe forward regions of the detector. Therefore in each detector proposal the forward regionsmust be shielded with massive masks, which will make difficult, or even impossible, taggingelectrons scattered at small angles. Electron tagging outside the mask, out of a cone ofabout 175 mrad will give access to a previously unexplored high Q2 range, Fig. 65 [156],but will give neither overlap with LEP2 results nor sensitivity to the small x region. Toachieve the overlap with LEP2 measurements, one needs an electron tagging device insidethe shielding, down to about 40 mrad, Fig. 65. This is however still not sufficient to unfoldstructure functions in the region x < 0.1, since small x corresponds to large W 2, where anunknown part of the hadronic system disappears undetected in the forward direction.

In order to circumevent this problem the eγ laser backscattering mode is ideal. Themethod to produce a beam of high energetic photons from an electron beam by means of theCompton backscattering process is shown in Fig. 66. The photons produced by high inten-sity laser are brought into collision with the electron beams at distances of about 0.1− 1 cmfrom the interaction point. The photons are scattered into a small cone around the initialelectron direction and receive a large fraction of the electron energy. Spent electrons aredeflected in a magnetic field. The invariant mass of the electron photon system is expectedto peak around

√seγ ≈ 0.8

√see [154].

The partonic content of the quasi-real and virtual photons has been also extensively stud-ied at the HERA collider mainly in photon-proton scattering and in deep inelastic electron-proton scattering. However the results obtained by the HERA experiments are beyond thescope of this review. The reader is refered to original publications or to review articlese.g. [6, 7, 11].

101

Acknowledgements

I would like to express my gratitude to Professor Danuta Kisielewska for her constant supportand encouragement since the time when I first met her as a student of physics at AGH.I would like to thank Richard Nisius and Albert de Roeck for the fruitful collaboration anduseful discussions on several topics presented in this paper.I am grateful to all the members of the OPAL Collaboration for discussions and friendlyatmosphere during my stay at CERN as a postdoctoral fellow.Last but not least, I would like to thank my parents for their support and understanding.

102

A Results on QED structure of the photon

In this Appendix we present the results on the QED photon structure functions F γ2,QED,

F γA,QED and F γ

B,QED obtained by the LEP experiments OPAL, DELPHI and L3 as well as bythe other experiments including CELLO, PLUTO and TPC/2γ. In all cases the structurefunctions scaled by the fine structure constant are given. Depending on the measurement,both statistical and systematic errors or only statistical or total error are given.Also the results of the measurement of the differential cross section dσ/dx for the scatteringof two virtual photons in the process e+e− → e+e− µ+µ− are listed.

CELLO Q2 = 1.2 − 39 GeV2 (9.5 GeV2)

x range 〈F γ2,QED〉/α ±σstat ±σsys

0.00 - 0.10 0.222 ±0.077 ±0.018

0.10 - 0.20 0.426 ±0.128 ±0.032

0.20 - 0.30 0.562 ±0.162 ±0.045

0.30 - 0.40 0.511 ±0.153 ±0.039

0.40 - 0.50 0.597 ±0.170 ±0.049

0.50 - 0.60 0.571 ±0.170 ±0.046

0.60 - 0.70 0.545 ±0.170 ±0.046

0.70 - 0.80 1.202 ±0.256 ±0.095

0.80 - 0.90 1.057 ±0.273 ±0.085

0.90 - 1.00 1.185 ±0.528 ±0.092

PLUTO Q2 = 1 − 16 GeV2 (5.5 GeV2)

x range 〈F γ2,QED〉/α ±σtot

0.00 - 0.10 0.081 ±0.040

0.10 - 0.20 0.177 ±0.048

0.20 - 0.30 0.532 ±0.089

0.30 - 0.40 0.403 ±0.105

0.40 - 0.50 0.532 ±0.113

0.50 - 0.60 0.597 ±0.161

0.60 - 0.70 0.952 ±0.322

0.70 - 0.80 0.887 ±0.444

TPC/2γ Q2 = 0.14 − 1.28 GeV2 (0.45 GeV2)

x range 〈F γ2,QED〉/α ±σstat ±σsys

0.00 - 0.05 0.038 ±0.010 ±0.007

0.05 - 0.10 0.104 ±0.010 ±0.017

0.10 - 0.15 0.135 ±0.017 ±0.021

0.15 - 0.20 0.172 ±0.021 ±0.028

0.20 - 0.25 0.219 ±0.031 ±0.035

0.25 - 0.30 0.281 ±0.042 ±0.044

0.30 - 0.35 0.320 ±0.042 ±0.051

0.35 - 0.40 0.344 ±0.037 ±0.055

0.40 - 0.50 0.370 ±0.042 ±0.059

0.50 - 0.60 0.373 ±0.037 ±0.059

0.60 - 0.70 0.357 ±0.037 ±0.057

0.70 - 0.80 0.354 ±0.037 ±0.057

0.80 - 0.90 0.291 ±0.031 ±0.047

0.90 - 1.00 0.323 ±0.068 ±0.051

PLUTO Q2 = 10 − 160 GeV2 (40 GeV2)

x range 〈F γ2,QED〉/α ±σtot

0.00 - 0.20 0.177 ±0.113

0.20 - 0.40 0.565 ±0.210

0.40 - 0.60 0.532 ±0.241

0.60 - 0.80 1.532 ±0.468

0.80 - 1.00 0.807 ±0.581

Table 14: The average photon structure function 〈F γ2,QED〉/α measured by the CELLO [114],

PLUTO [115] and TPC/2γ [116] experiments. The numbers listed here are in all cases readoff the published figures. For the consistency they are the same as given in [7]. The Q2 rangeof each measurement, with the average value in parentheses, is given next to the experimentname. No information is avalible on the average 〈P 2〉 in those measurements.

103

OPAL Q2 = 1.5 − 3 GeV2 (2.2 GeV2)

x range F γ2,QED/α ±σstat ±σsys

0.00 - 0.10 0.115 ±0.007 ±0.005

0.10 - 0.20 0.219 ±0.010 ±0.008

0.20 - 0.30 0.282 ±0.012 ±0.011

0.30 - 0.40 0.347 ±0.015 ±0.011

0.40 - 0.50 0.356 ±0.017 ±0.010

0.50 - 0.60 0.400 ±0.020 ±0.011

0.60 - 0.70 0.483 ±0.025 ±0.016

0.70 - 0.80 0.491 ±0.031 ±0.012

0.80 - 0.90 0.532 ±0.034 ±0.013

0.90 - 0.97 0.308 ±0.032 ±0.071

OPAL Q2 = 6 − 10 GeV2 (8.4 GeV2)

x range F γ2,QED/α ±σstat ±σsys

0.00 - 0.10 0.090 ±0.012 ±0.007

0.10 - 0.20 0.271 ±0.022 ±0.019

0.20 - 0.30 0.334 ±0.029 ±0.020

0.30 - 0.40 0.409 ±0.033 ±0.022

0.40 - 0.50 0.496 ±0.038 ±0.026

0.50 - 0.60 0.563 ±0.043 ±0.026

0.60 - 0.70 0.596 ±0.049 ±0.023

0.70 - 0.80 0.687 ±0.056 ±0.023

0.80 - 0.90 0.891 ±0.072 ±0.044

0.90 - 0.97 0.761 ±0.074 ±0.049

OPAL Q2 = 15 − 30 GeV2 (21 GeV2)

x range F γ2,QED/α ±σstat ±σsys

0.00 - 0.15 0.117 ±0.028 ±0.012

0.15 - 0.30 0.302 ±0.039 ±0.021

0.30 - 0.45 0.403 ±0.051 ±0.029

0.45 - 0.60 0.559 ±0.058 ±0.030

0.60 - 0.75 0.782 ±0.070 ±0.034

0.75 - 0.90 0.907 ±0.080 ±0.033

0.90 - 0.97 0.802 ±0.103 ±0.033

OPAL Q2 = 3 − 7 GeV2 (4.2 GeV2)

x range F γ2,QED/α ±σstat ±σsys

0.00 - 0.10 0.108 ±0.010 ±0.016

0.10 - 0.20 0.237 ±0.014 ±0.009

0.20 - 0.30 0.320 ±0.018 ±0.012

0.30 - 0.40 0.378 ±0.020 ±0.011

0.40 - 0.50 0.373 ±0.020 ±0.010

0.50 - 0.60 0.421 ±0.025 ±0.012

0.60 - 0.70 0.519 ±0.029 ±0.013

0.70 - 0.80 0.556 ±0.034 ±0.011

0.80 - 0.90 0.601 ±0.040 ±0.013

0.90 - 0.97 0.470 ±0.041 ±0.051

OPAL Q2 = 10 − 15 GeV2 (12.4 GeV2)

x range F γ2,QED/α ±σstat ±σsys

0.00 - 0.10 0.095 ±0.010 ±0.007

0.10 - 0.20 0.264 ±0.018 ±0.014

0.20 - 0.30 0.319 ±0.023 ±0.014

0.30 - 0.40 0.428 ±0.028 ±0.020

0.40 - 0.50 0.446 ±0.030 ±0.018

0.50 - 0.60 0.558 ±0.034 ±0.023

0.60 - 0.70 0.698 ±0.040 ±0.030

0.70 - 0.80 0.770 ±0.044 ±0.026

0.80 - 0.90 0.871 ±0.053 ±0.033

0.90 - 0.97 0.795 ±0.053 ±0.044

OPAL Q2 = 70 − 400 GeV2 (130 GeV2)

x range F γ2,QED/α ±σstat ±σsys

0.00 - 0.40 0.343 ±0.094 ±0.034

0.40 - 0.60 0.578 ±0.079 ±0.052

0.60 - 0.80 0.936 ±0.109 ±0.063

0.80 - 0.90 1.125 ±0.130 ±0.057

Table 15: The photon structure function F γ2,QED/α measured by the OPAL experiment [111].

The Q2 range of each measurement, with the average value in parentheses, is given next tothe experiment name. The average virtuality of the target photon in all the measurementsis 〈P 2〉 = 0.05 GeV2.

104

DELPHI Q2 = 2.4 − 51.2 GeV2 (12.5 GeV2)

x range 〈F γ2,QED〉/α ±σstat ±σsys

0.00 - 0.10 0.106 ±0.008 ±0.023

0.10 - 0.20 0.273 ±0.012 ±0.012

0.20 - 0.30 0.426 ±0.017 ±0.012

0.30 - 0.40 0.515 ±0.021 ±0.012

0.40 - 0.50 0.573 ±0.024 ±0.004

0.50 - 0.60 0.645 ±0.029 ±0.003

0.60 - 0.70 0.743 ±0.038 ±0.021

0.70 - 0.80 0.942 ±0.060 ±0.053

0.80 - 1.00 1.152 ±0.112 ±0.094

DELPHI Q2 = 45.9 − 752.8 GeV2 (120 GeV2)

x range 〈F γ2,QED〉/α ±σstat ±σsys

0.00 - 0.20 0.387 ±0.214 ±0.015

0.20 - 0.40 0.464 ±0.133 ±0.051

0.40 - 0.60 0.673 ±0.138 ±0.049

0.60 - 0.80 0.984 ±0.162 ±0.026

0.80 - 1.00 1.508 ±0.231 ±0.044

L3 Q2 = 1.4 − 7.6 GeV2 (3.25 GeV2)

x range 〈F γ2,QED〉/α ±σstat ±σsys

0.00 - 0.10 0.062 ±0.006 ±0.002

0.10 - 0.20 0.216 ±0.015 ±0.007

0.20 - 0.30 0.326 ±0.022 ±0.011

0.30 - 0.40 0.391 ±0.026 ±0.013

0.40 - 0.50 0.477 ±0.028 ±0.016

0.50 - 0.60 0.534 ±0.029 ±0.018

0.60 - 0.70 0.654 ±0.035 ±0.022

0.70 - 0.80 0.709 ±0.037 ±0.023

0.80 - 0.90 0.775 ±0.046 ±0.026

0.90 - 1.00 0.549 ±0.069 ±0.018

Table 16: The average photon structure function 〈F γ2,QED〉/α measured by the DELPHI [112]

and by the L3 [113] experiments. The Q2 range of each measurement, with the average valuein parentheses, is given next to the experiment name. The average virtuality of the targetphoton in the DELPHI measurement is 〈P 2〉 = 0.025±0.005 GeV2 and 〈P 2〉 = 0.073±0.056GeV2 for the samples with low and high Q2, respectively, and in the L3 measurement 〈P 2〉 =0.033 GeV2.

OPAL Q2 = 1.5 − 30 GeV2 〈Q2〉 = 5.4 GeV2

x range F γ2,QED/α ±σstat ±σsys F γ

A,QED/α ±σstat ±σsys F γB,QED/α ±σstat ±σsys

x < 0.25 0.249 ±0.006 ±0.008 0.039 ±0.007 ±0.003 0.029 ±0.010 ±0.003

0.25 − 0.50 0.523 ±0.011 ±0.014 0.011 ±0.016 ±0.004 0.101 ±0.025 ±0.011

0.50 − 0.75 0.738 ±0.017 ±0.019 −0.122 ±0.021 ±0.006 0.121 ±0.041 ±0.017

0.75 < x 0.871 ±0.027 ±0.021 −0.201 ±0.033 ±0.013 0.063 ±0.056 ±0.018

L3 Q2 = 1.4 − 7.6 GeV2 〈Q2〉 = 3.35 GeV2

x range F γ2,QED/α ±σtot F γ

A,QED/α ±σtot F γB,QED/α ±σtot

0.00 − 0.25 0.090 ±0.008 0.014 ±0.024 0.008 ±0.010

0.25 − 0.50 0.404 ±0.016 0.036 ±0.032 0.090 ±0.021

0.50 − 0.75 0.597 ±0.020 −0.126 ±0.052 0.168 ±0.040

0.75 − 1.00 0.731 ±0.032 −0.174 ±0.062 0.089 ±0.045

Table 17: The photon structure functions F γA,QED/α and F γ

B,QED/α measured by the OPAL[111] and L3 [113] experiments. Due to sligthly diffeent definition of F γ

A,QED in the L3publication, the original numbers for F γ

A,QED given in Table 2 in [113] have been multipliedby −2.

105

OPAL Q2 = 1.5 − 30 GeV2 〈Q2〉 = 5.4 GeV2

x range F γA,QED/F γ

2,QED ±σstat ±σsys12F γ

B,QED/F γ2,QED ±σstat ±σsys

x < 0.25 0.176 ±0.031 ±0.010 0.075 ±0.025 ±0.008

0.25 − 0.50 0.018 ±0.028 ±0.008 0.099 ±0.024 ±0.010

0.50 − 0.75 −0.171 ±0.029 ±0.007 0.081 ±0.027 ±0.011

0.75 < x -0.228 ±0.037 ±0.014 0.037 ±0.033 ±0.011

L3 Q2 = 1.4 − 7.6 GeV2 〈Q2〉 = 3.35 GeV2

x range F γA,QED/F γ

2,QED ±σstat ±σsys12F γ

B,QED/F γ2,QED ±σstat ±σsys

0.00 − 0.25 0.159 ±0.040 ±0.034 0.046 ±0.012 ±0.012

0.25 − 0.50 0.087 ±0.071 ±0.056 0.111 ±0.019 ±0.038

0.50 − 0.75 −0.210 ±0.102 ±0.057 0.141 ±0.026 ±0.048

0.75 − 1.00 −0.236 ±0.091 ±0.079 0.061 ±0.019 ±0.030

DELPHI Q2 = 2.4 − 51.2 GeV2 〈Q2〉 = 12.5 GeV2

x range F γA,QED/F γ

2,QED ±σtot12F γ

B,QED/F γ2,QED ±σtot

x < 0.20 0.135 ±0.043 0.004 ±0.027

0.20 − 0.40 0.140 ±0.035 0.077 ±0.026

0.40 − 0.60 0.038 ±0.044 0.099 ±0.035

0.60 < x −0.263 ±0.059 0.182 ±0.040

Table 18: The photon structure function ratios F γA,QED/F γ

2,QED and 12F γ

B,QED/F γ2,QED mea-

sured by the OPAL [111], L3 [113] and DELPHI [112] experiments. Due to sligthly diffeentdefinition of F γ

A,QED in the L3 publication, the original numbers for F γA,QED given in Table 2

in [113] have been multiplied by −2. In case of DELPHI the total error given is a sum inquadrature of statistical and systematic errors.

OPAL

〈Q2〉 = 3.6 GeV2 〈P 2〉 = 2.3 GeV2

x range dσ/dx ±σstat ±σsys

0.00 - 0.20 9.77 ±1.62 ±1.80

0.20 - 0.40 10.45 ±1.26 ±1.39

0.40 - 0.65 4.34 ±1.07 ±1.09

OPAL

〈Q2〉 = 14 GeV2 〈P 2〉 = 5 GeV2

x range dσ/dx ±σstat ±σsys

0.00 - 0.25 5.26 ±0.82 ±1.29

0.25 - 0.50 6.87 ±0.78 ±1.08

0.50 - 0.75 2.75 ±0.60 ±0.63

Table 19: The cross section for the scattering of two virtual photons in the process e+e− →e+e−µ+µ− measured by the OPAL experiment [111].

106

B Results on hadronic structure of the photon

In this Appendix we present the results on the hadronic photon structure function F γ2 ob-

tained by the LEP experiments OPAL, ALEPH, DELPHI and L3 as well as by the otherexperiments including AMY, CELLO, JADE, PLUTO, TASSO, TOPAZ and TPC/2γ. Inall cases the structure functions scaled by the fine structure constant are given. Dependingon the measurement, both statistical and systematic errors or only statistical or total errorare given.

107

AMY

Q2 = 3.5 − 12 GeV2 〈Q2〉 = 6.8 GeV2

x range F γ2 ±σstat ±σsys

0.015 - 0.125 0.337 ±0.030 ±0.044

0.125 - 0.375 0.302 ±0.040 ±0.029

0.375 - 0.620 0.322 ±0.049 ±0.084

Q2 = 25 − 220 GeV2 〈Q2〉 = 73 GeV2

x range F γ2 ±σstat ±σsys

0.125 - 0.375 0.65 ±0.08 ±0.06

0.375 - 0.625 0.60 ±0.16 ±0.03

0.625 - 0.875 0.65 ±0.11 ±0.08

Q2 > 110 GeV2 〈Q2〉 = 390 GeV2

x range F γ2 ±σstat ±σsys

0.120 - 0.500 0.94 ±0.23 ±0.10

0.500 - 0.800 0.82 ±0.16 ±0.11

JADE

Q2 = 10 − 55 GeV2 〈Q2〉 = 24 GeV2

x range F γ2 ±σtot

0.000 - 0.100 0.51 ±0.15

0.100 - 0.200 0.29 ±0.12

0.200 - 0.400 0.34 ±0.10

0.400 - 0.600 0.59 ±0.12

0.600 - 0.900 0.23 ±0.12

Q2 = 30 − 220 GeV2 〈Q2〉 = 100 GeV2

x range F γ2 ±σtot

0.100 - 0.300 0.52 ±0.23

0.300 - 0.600 0.75 ±0.22

0.600 - 0.900 0.90 ±0.27

TOPAZ

Q2 = 3 − 10 GeV2 〈Q2〉 = 5.1 GeV2

x range F γ2 /α ±σstat ±σsys

0.010 - 0.076 0.33 ±0.02 ±0.05

0.076 - 0.200 0.29 ±0.03 ±0.03

Q2 = 10 − 30 GeV2 〈Q2〉 = 16 GeV2

x range 〈F γ2 〉/α ±σstat ±σsys

0.020 - 0.150 0.60 ±0.08 ±0.06

0.150 - 0.330 0.56 ±0.09 ±0.04

0.330 - 0.780 0.46 ±0.15 ±0.06

Q2 = 45 − 130 GeV2 〈Q2〉 = 80 GeV2

x range 〈F γ2 〉/α ±σstat ±σsys

0.060 - 0.320 0.68 ±0.26 ±0.05

0.320 - 0.590 0.83 ±0.22 ±0.05

0.590 - 0.980 0.53 ±0.21 ±0.05

TASSO

Q2 = 7 − 70 GeV2 〈Q2〉 = 23 GeV2

x range F γ2 /α ±σtot

0.020 - 0.200 0.366 ±0.089

0.200 - 0.400 0.670 ±0.086

0.400 - 0.600 0.722 ±0.104

0.600 - 0.800 0.693 ±0.116

0.800 - 0.980 0.407 ±0.222

Table 20: The photon structure function F γ2 /α measured by the AMY [123,124], JADE [125],

TOPAZ [130] and TASSO [128] experiments. The Q2 range of each measurement, with theaverage value in parentheses, is given next to the experiment name. The quoted numbers forthe JADE and TASSO measurements have been taken from the RAL database, which wereread off the figures in the original publications and contain probably only the total errors.

108

TPC/2γ

Q2 = 0.2 − 0.3 GeV2 〈Q2〉 = 0.24 GeV2

x range F γ2 /α ±σstat ±σsys

0.000 - 0.020 0.084 ±0.005 ±0.001

0.020 - 0.060 0.074 ±0.008 ±0.001

0.060 - 0.180 0.062 ±0.013 ±0.001

Q2 = 0.3 − 0.5 GeV2 〈Q2〉 = 0.38 GeV2

x range F γ2 /α ±σstat ±σsys

0.000 - 0.020 0.113 ±0.007 ±0.001

0.020 - 0.055 0.118 ±0.011 ±0.001

0.055 - 0.111 0.171 ±0.021 ±0.002

0.111 - 0.243 0.151 ±0.028 ±0.002

Q2 = 0.5 − 1 GeV2 〈Q2〉 = 0.71 GeV2

x range F γ2 /α ±σstat ±σsys

0.000 - 0.028 0.117 ±0.006 ±0.001

0.028 - 0.065 0.130 ±0.010 ±0.001

0.065 - 0.121 0.170 ±0.017 ±0.002

0.121 - 0.340 0.133 ±0.013 ±0.002

PLUTO

Q2 = 1.5 − 3 GeV2 〈Q2〉 = 2.4 GeV2

x range F γ2 /α ±σstat ±σsys

0.016 - 0.110 0.183 ±0.014 ±0.046

0.110 - 0.370 0.263 ±0.026 ±0.039

0.370 - 0.700 0.222 ±0.064 ±0.033

Q2 = 3 − 6 GeV2 〈Q2〉 = 4.3 GeV2

x range F γ2 /α ±σstat ±σsys

0.030 - 0.170 0.218 ±0.014 ±0.055

0.170 - 0.440 0.273 ±0.020 ±0.041

0.440 - 0.800 0.336 ±0.044 ±0.050

Q2 = 6 − 16 GeV2 〈Q2〉 = 9.2 GeV2

x range F γ2 /α ±σstat ±σsys

0.060 - 0.230 0.300 ±0.027 ±0.075

0.230 - 0.540 0.340 ±0.029 ±0.051

0.540 - 0.900 0.492 ±0.069 ±0.074

TPC/2γ

Q2 = 1 − 1.6 GeV2 〈Q2〉 = 1.3 GeV2

x range F γ2 /α ±σstat ±σsys

0.000 - 0.050 0.107 ±0.013 ±0.001

0.050 - 0.126 0.184 ±0.021 ±0.002

0.126 - 0.215 0.215 ±0.034 ±0.003

0.215 - 0.507 0.102 ±0.031 ±0.002

Q2 = 1.8 − 4 GeV2 〈Q2〉 = 2.8 GeV2

x range F γ2 /α ±σstat ±σsys

0.000 - 0.080 0.134 ±0.018 ±0.002

0.080 - 0.156 0.234 ±0.031 ±0.003

0.156 - 0.303 0.198 ±0.042 ±0.002

0.303 - 0.600 0.160 ±0.033 ±0.002

Q2 = 4 − 6.6 GeV2 〈Q2〉 = 5.1 GeV2

x range F γ2 /α ±σstat ±σsys

0.021 - 0.199 0.224 ±0.034 ±0.003

0.199 - 0.359 0.373 ±0.057 ±0.005

0.359 - 0.740 0.300 ±0.044 ±0.004

PLUTO

Q2 = 18 − 100 GeV2 〈Q2〉 = 45 GeV2

x range F γ2 /α ±σstat ±σsys

0.100 - 0.250 0.360 ±0.170 ±0.036

0.250 - 0.500 0.400 ±0.120 ±0.040

0.500 - 0.750 0.770 ±0.160 ±0.077

0.750 - 0.900 0.840 ±0.260 ±0.084

Q2 = 1.5 − 16 GeV2 〈Q2〉 = 5.3 GeV2

x range F γ2 /α ±σstat ±σsys

0.035 - 0.072 0.216 ±0.015 ±0.054

0.072 - 0.174 0.258 ±0.010 ±0.065

0.174 - 0.319 0.222 ±0.025 ±0.056

0.319 - 0.490 0.329 ±0.037 ±0.049

0.490 - 0.650 0.439 ±0.052 ±0.066

0.650 - 0.840 0.361 ±0.076 ±0.054

Table 21: The photon structure function F γ2 measured by the TPC/2γ [129] and PLUTO

[126,127] experiments. The Q2 range of each measurement, with the average value in paren-theses, is given next to the experiment name. The quoted numbers for F γ

2 /α with thestatistical errors have been taken from the RAL database. The quoted systematic errorsin case of TPC/2γ measurements amount to 11% for the regions 0.2 < Q2 < 1 GeV2 withx < 0.1 and 1 < Q2 < 7 GeV2 with x < 0.2, and to 14% elsewhere [129], and in case ofPLUTO the systematic error amounts to 10% for data at 〈Q2〉 = 45 GeV62 [127], and to25/15% for data at lower 〈Q2〉 and for below/above x = 0.2, respectively [126].

109

OPAL

Q2 = 6 − 8 GeV2 〈Q2〉 = 7.5 GeV2

x range F γ2 /α ±σstat ±σsys

0.001 - 0.091 0.28 ±0.02 +0.03−0.10

0.091 - 0.283 0.32 ±0.02 +0.08−0.13

0.283 - 0.649 0.38 ±0.04 +0.06−0.21

Q2 = 8 − 30 GeV2 〈Q2〉 = 14.7 GeV2

x range F γ2 /α ±σstat ±σsys

0.006 - 0.137 0.38 ±0.01 +0.06−0.13

0.137 - 0.324 0.41 ±0.02 +0.06−0.03

0.324 - 0.522 0.41 ±0.03 +0.08−0.11

0.522 - 0.836 0.54 ±0.05 +0.31−0.13

Q2 = 60 − 400 GeV2 〈Q2〉 = 135 GeV2

x range F γ2 /α ±σstat ±σsys

0.100 - 0.300 0.65 ±0.09 +0.33−0.06

0.300 - 0.600 0.73 ±0.08 +0.04−0.08

0.600 - 0.800 1.72 ±0.10 +0.81−0.07

Q2 = 6 − 11 GeV2 〈Q2〉 = 9.0 GeV2

x range F γ2 /α ±σstat ±σsys

0.020 - 0.100 0.33 ±0.03 +0.06−0.06

0.100 - 0.250 0.29 ±0.04 +0.04−0.05

0.250 - 0.600 0.39 ±0.08 +0.30−0.10

Q2 = 11 − 20 GeV2 〈Q2〉 = 14.5 GeV2

x range F γ2 /α ±σstat ±σsys

0.020 - 0.100 0.37 ±0.03 +0.16−0.01

0.100 - 0.250 0.42 ±0.05 +0.04−0.14

0.250 - 0.600 0.39 ±0.06 +0.10−0.11

Q2 = 20 − 40 GeV2 〈Q2〉 = 30 GeV2

x range F γ2 /α ±σstat ±σsys

0.050 - 0.100 0.32 ±0.04 +0.11−0.02

0.100 - 0.350 0.52 ±0.05 +0.06−0.13

0.350 - 0.600 0.41 ±0.09 +0.20−0.05

0.600 - 0.800 0.46 ±0.15 +0.39−0.14

OPAL

Q2 = 40 − 100 GeV2 〈Q2〉 = 59 GeV2

x range F γ2 /α ±σstat ±σsys

0.050 - 0.100 0.37 ±0.06 +0.28−0.07

0.100 - 0.350 0.44 ±0.07 +0.08−0.07

0.350 - 0.600 0.48 ±0.09 +0.16−0.10

0.600 - 0.800 0.51 ±0.14 +0.48−0.02

Q2 = 1.1 − 2.5 GeV2 〈Q2〉 = 1.86 GeV2

x range F γ2 /α ±σstat ±σsys

0.0025 - 0.0063 0.27 ±0.03 +0.05−0.07

0.0063 - 0.0200 0.22 ±0.02 +0.02−0.05

0.0200 - 0.0400 0.20 ±0.02 +0.09−0.02

0.0400 - 0.1000 0.23 ±0.02 +0.03−0.05

Q2 = 2.5 − 6.6 GeV2 〈Q2〉 = 3.76 GeV2

x range F γ2 /α ±σstat ±σsys

0.0063 - 0.0200 0.35 ±0.03 +0.08−0.08

0.0200 - 0.0400 0.29 ±0.03 +0.06−0.06

0.0400 - 0.1000 0.32 ±0.02 +0.07−0.05

0.1000 - 0.2000 0.32 ±0.03 +0.08−0.04

Q2 = 400 − 2350 GeV2 〈Q2〉 = 780 GeV2

x range F γ2 /α ±σstat ±σsys

0.15 - 0.40 0.93 ±0.10 +0.14−0.11

0.40 - 0.70 0.87 ±0.10 +0.05−0.15

0.70 - 0.98 0.97 ±0.17 +0.16−0.23

Q2 ≃ 7 − 14 GeV2 〈Q2〉 = 10.2 GeV2

x range F γ2 /α ±σstat ±σsys

0.0009 - 0.0050 0.545 ±0.016 +0.066−0.071

0.0050 - 0.0273 0.349 ±0.007 +0.030−0.033

0.0273 - 0.1496 0.315 ±0.006 +0.023−0.024

0.1496 - 0.8187 0.417 ±0.008 +0.052−0.052

Q2 ≃ 14 − 33 GeV2 〈Q2〉 = 20 GeV2

x range F γ2 /α ±σstat ±σsys

0.0015 - 0.0074 0.626 ±0.019 +0.123−0.129

0.0074 - 0.0369 0.395 ±0.008 +0.047−0.051

0.0369 - 0.1827 0.380 ±0.006 +0.019−0.022

0.1827 - 0.9048 0.500 ±0.008 +0.067−0.066

Table 22: The photon structure function F γ2 measured by the OPAL experiment [131, 132].

The Q2 range of each measurement, with the average value in parentheses, is given next tothe experiment name. The quoted numbers for F γ

2 /α with the statistical and systematicerrors have been taken from the published tables.

110

ALEPH

Q2 = 6 − 13 GeV2 〈Q2〉 = 9.9 GeV2

x range F γ2 /α ±σstat ±σsys

0.005 - 0.080 0.30 ±0.02 ±0.02

0.080 - 0.200 0.40 ±0.03 ±0.07

0.200 - 0.400 0.41 ±0.05 ±0.09

0.400 - 0.800 0.27 ±0.13 ±0.09

Q2 = 13 − 44 GeV2 〈Q2〉 = 20.7 GeV2

x range F γ2 /α ±σstat ±σsys

0.009 - 0.120 0.36 ±0.02 ±0.05

0.120 - 0.270 0.34 ±0.03 ±0.11

0.270 - 0.500 0.56 ±0.05 ±0.10

0.350 - 0.890 0.45 ±0.11 ±0.05

Q2 = 35 − 3000 GeV2 〈Q2〉 = 284 GeV2

x range F γ2 /α ±σstat ±σsys

0.03 - 0.35 0.65 ±0.10 ±0.09

0.35 - 0.65 0.70 ±0.16 ±0.19

0.65 - 0.97 1.28 ±0.26 ±0.26

ALEPH

Q2 ≃ 10 − 29 GeV2 〈Q2〉 = 17.3 GeV2

x range F γ2 /α ±σstat ±σsys

0.0020 - 0.0110 0.43 ±0.016 ±0.115

0.0110 - 0.0338 0.27 ±0.014 ±0.049

0.0338 - 0.0787 0.35 ±0.015 ±0.041

0.0787 - 0.1487 0.35 ±0.015 ±0.028

0.1487 - 0.2429 0.39 ±0.016 ±0.034

0.2429 - 0.3624 0.46 ±0.018 ±0.041

0.3624 - 0.5074 0.40 ±0.021 ±0.149

0.5074 - 0.7000 0.18 ±0.019 ±0.235

Q2 ≃ 29 − 250 GeV2 〈Q2〉 = 67.2 GeV2

x range F γ2 /α ±σstat ±σsys

0.0060 - 0.0362 0.57 ±0.027 ±0.145

0.0362 - 0.0950 0.43 ±0.027 ±0.039

0.0950 - 0.1811 0.47 ±0.029 ±0.041

0.1811 - 0.2907 0.50 ±0.031 ±0.031

0.2907 - 0.4204 0.60 ±0.036 ±0.038

0.4204 - 0.5714 0.66 ±0.038 ±0.077

0.5714 - 0.7356 0.65 ±0.055 ±0.126

0.7356 - 0.9600 0.66 ±0.060 ±0.137

Table 23: The photon structure function F γ2 measured by the ALEPH experiment [131,132].

The Q2 range of each measurement, with the average value in parentheses, is given next tothe experiment name. The quoted numbers for F γ

2 /α with the statistical and systematicerrors have been taken from the published tables.

111

DELPHI

Q2 = 4 − 30 GeV2 〈Q2〉 = 12 GeV2

x range F γ2 /α ±σstat ±σsys

0.003 - 0.080 0.21 ±0.03 ±0.06

0.080 - 0.213 0.41 ±0.04 ±0.05

0.213 - 0.428 0.45 ±0.05 ±0.05

0.428 - 0.847 0.45 ±0.11 ±0.10

DELPHI

Q2 = ....... GeV2 〈Q2〉 = 5.2 GeV2

x range F γ2 /α ±σstat ±σsys

0.001 - 0.020 0.283 ±0.011 ±0.026

0.020 - 0.100 0.203 ±0.014 ±0.024

0.100 - 0.500 0.212 ±0.020 ±0.033

Q2 = ...... GeV2 〈Q2〉 = 12.7 GeV2

x range F γ2 /α ±σstat ±σsys

0.001 - 0.020 0.446 ±0.015 ±0.027

0.020 - 0.100 0.317 ±0.018 ±0.013

0.100 - 0.300 0.286 ±0.034 ±0.015

0.300 - 0.800 0.362 ±0.015 ±0.029

Q2 = ...... GeV2 〈Q2〉 = 28.5 GeV2

x range F γ2 /α ±σstat ±σsys

0.020 - 0.100 0.413 ±0.030 ±0.035

0.100 - 0.300 0.346 ±0.015 ±0.034

0.300 - 0.800 0.488 ±0.037 ±0.033

DELPHI

Q2 = 4 − 30 GeV2 〈Q2〉 = 12 GeV2

x range F γ2 /α ±σstat ±σsys

0.003 - 0.046 0.24 ±0.03 ±0.07

0.046 - 0.117 0.41 ±0.05 ±0.08

0.117 - 0.350 0.46 ±0.17 ±0.09

DELPHI

Q2 = ...... GeV2 〈Q2〉 = 19 GeV2

x range F γ2 /α ±σstat ±σsys

0.001 - 0.020 0.438 ±0.013 ±0.063

0.020 - 0.100 0.317 ±0.010 ±0.029

0.100 - 0.300 0.293 ±0.007 ±0.061

0.300 - 0.800 0.341 ±0.026 ±0.042

Q2 = ...... GeV2 〈Q2〉 = 40 GeV2

x range F γ2 /α ±σstat ±σsys

0.001 - 0.020 0.632 ±0.010 ±0.051

0.020 - 0.100 0.428 ±0.007 ±0.064

0.100 - 0.300 0.351 ±0.018 ±0.092

0.300 - 0.800 0.487 ±0.017 ±0.119

Q2 = ...... GeV2 〈Q2〉 = 101 GeV2

x range F γ2 /α ±σstat ±σsys

0.001 - 0.020 0.930 ±0.106 ±0.131

0.020 - 0.100 0.585 ±0.034 ±0.084

0.100 - 0.300 0.425 ±0.020 ±0.101

0.300 - 0.800 0.704 ±0.031 ±0.094

Q2 = ...... GeV2 〈Q2〉 = 700 GeV2

x range F γ2 /α ±σstat ±σsys

0.010 - 0.300 0.797 ±0.090 ±0.226

0.300 - 0.800 0.962 ±0.052 ±0.111

Table 24: The photon structure function F γ2 measured by the DELPHI experiment [131,132].

The two measurements at 〈Q2〉 = 12 GeV2 are not independent, but use the same data whichare unfolded for four bins on a linear scale in x, and for three bins on a logarithmic scalefor x < 0.35. The quoted numbers for F γ

2 /α with the statistical and systematic errors havebeen taken from the published tables.

112

L3

Q2 = 1.2 − 3 GeV2 〈Q2〉 = 1.9 GeV2

x range F γ2 ±σstat ±σsys

0.002 - 0.005 0.184 ±0.009 ±0.049

0.005 - 0.010 0.179 ±0.007 ±0.022

0.010 - 0.020 0.176 ±0.006 ±0.016

0.020 - 0.030 0.191 ±0.008 ±0.005

0.030 - 0.050 0.193 ±0.008 ±0.009

0.050 - 0.100 0.185 ±0.007 ±0.026

Q2 = 3 − 9 GeV2 〈Q2〉 = 5 GeV2

x range F γ2 ±σstat ±σsys

0.005 - 0.010 0.307 ±0.021 ±0.094

0.010 - 0.020 0.282 ±0.014 ±0.045

0.020 - 0.040 0.263 ±0.011 ±0.021

0.040 - 0.060 0.278 ±0.013 ±0.007

0.060 - 0.100 0.270 ±0.012 ±0.009

0.100 - 0.200 0.252 ±0.011 ±0.045

Q2 = 9 − 13 GeV2 〈Q2〉 = 10.8 GeV2

x range F γ2 ±σstat ±σsys

0.01 - 0.10 0.30 ±0.02 ±0.03

0.10 - 0.20 0.35 ±0.03 ±0.02

0.20 - 0.30 0.30 ±0.04 ±0.11

L3

Q2 = 13 − 18 GeV2 〈Q2〉 = 15.3 GeV2

x range F γ2 ±σstat ±σsys

0.01 - 0.10 0.37 ±0.02 ±0.03

0.10 - 0.20 0.42 ±0.04 ±0.02

0.20 - 0.30 0.42 ±0.05 ±0.07

0.30 - 0.50 0.35 ±0.05 ±0.15

Q2 = 18 − 30 GeV2 〈Q2〉 = 23.1 GeV2

x range F γ2 ±σstat ±σsys

0.01 - 0.10 0.40 ±0.03 ±0.04

0.10 - 0.20 0.44 ±0.04 ±0.04

0.20 - 0.30 0.47 ±0.05 ±0.02

0.30 - 0.50 0.44 ±0.05 ±0.12

L3

Q2 = 40 − 500 GeV2 〈Q2〉 = 120 GeV2

x range F γ2 /α ±σstat ±σsys

0.05 - 0.20 0.66 ±0.08 ±0.06

0.20 - 0.40 0.81 ±0.08 ±0.08

0.40 - 0.60 0.76 ±0.12 ±0.07

0.60 - 0.80 0.85 ±0.14 ±0.08

0.80 - 0.98 0.91 ±0.19 ±0.09

Table 25: The photon structure function F γ2 measured by the L3 experiment [135,136,146].

The Q2 range of each measurement, with the average value in parentheses, is given next tothe experiment name. In the published tables two sets of numbers are given, one obtained byan unfolding based on PHOJET Monte Carlo and the other based on the TWOGAM MonteCarlo. The numbers quoted numbers here for F γ

2 /α are those obtained from PHOJET withthe statistical errors. The systematic error is calculated from the quadratic sum of thesystematic error for the result based on PHOJET and the difference bbetween the resultsobtained from PHOJET and from TWOGAM.

OPAL 〈Q2〉 = 20 GeV2

x range σD⋆

[pb] σ(e+e− → e+e−cc X) [pb] F γ2,c/α

0.0014 - 0.10 3.1 ± 1.0 ± 0.5 43.8 ± 14.3 ± 6.3 ± 2.8 0.180 ± 0.059 ± 0.026 ± 0.012

0.1000 - 0.87 2.6 ± 0.9 ± 0.3 26.2 ± 8.8 ± 3.2 ± 1.3 0.084 ± 0.028 ± 0.010 ± 0.004

Table 26: Measurement of the cross section σD⋆

in the restricted region, the cross sectionσ(e+e− → e+e−cc X, extrapolated using Monte Carlo and the structure function F γ

2,c/α atQ2 = 20 GeV2 by the OPAL experiment [142]. For the measurements the quoted errors aredue, respectively, statistics, systematics and extrapolation.

113

C Results on hadronic structure of the electron

In this Appendix we present the results on the hadronic structure function of the electronF e

2 measured by the OPAL and DELPHI experiments. In all cases the structure functionscaled by the fine structure constant squared is given.

OPAL 〈Q2〉 = 15 GeV2

z

range bin-centredσ/dz [pb] F e

2 /α2

0.0009 - 0.00265 0.00154 6141 ± 57 +525−533 13.14 ± 0.12 +1.21

−1.07

0.00265 - 0.0078 0.00455 2996 ± 22 +279−280 10.04 ± 0.08 +0.90

−0.81

0.0078 - 0.0230 0.01339 857 ± 8 +34−48 7.84 ± 0.07 +0.45

−0.53

0.0230 - 0.0677 0.03946 221 ± 3 +30−28 5.69 ± 0.09 +0.60

−0.63

0.0677 - 0.2000 0.11636 48 ± 1 +9−7 3.43 ± 0.10 +0.60

−0.56

0.2000 - 0.9048 0.42539 2.8 ± 0.2 +1.2−0.6 0.71 ± 0.04 +0.23

−0.15

Table 27: Results for the cross section dσ/dz obtaineded in the phase space defined byE ′ > 0.75Eb, 34 < θ < 55 mrad and W >

√3 GeV and the structure function F e

2 /α2 forthe average 〈Q2〉 = 15 GeV2 measured by the OPAL experiment [143]. The first errors arestatistical and the second systematic.

DELPHI 〈Q2〉 = 8.9 GeV2

z range F e2 /α2

0.00398 - 0.01000 6.98 +0.41−0.43

0.01000 - 0.02512 5.45 +0.39−0.42

0.02512 - 0.06310 3.58 +0.52−0.49

0.06310 - 0.15849 1.77 +0.59−0.62

0.15849 - 0.39811 0.21 +0.65−0.71

Table 28: Results for F e2 /α2 obtained for the average 〈Q2〉 = 8.9 GeV2 measured by the

DELPHI experiment [144] at LEP1 energy. Only the total erro is available. There is noerror due to model dependence.

114

DELPHI 〈Q2〉 = 15.3 GeV2

z range F e2 /α2

0.00151 - 0.00204 12.06 ± 1.68 +1.59−0.93

0.00204 - 0.00275 9.92 ± 0.98 +0.90−0.93

0.00275 - 0.00372 10.52 ± 0.96 +1.07−0.93

0.00372 - 0.00500 7.06 ± 1.12 +0.61−0.93

DELPHI 〈Q2〉 = 24.5 GeV2

z range F e2 /α2

0.00295 - 0.00575 10.54 ± 0.75 +0.70−0.96

0.00575 - 0.01122 8.98 ± 0.50 +0.57−0.62

0.01122 - 0.02188 7.05 ± 0.48 +0.56−0.66

0.02188 - 0.04266 5.74 ± 0.62 +0.53−0.59

0.04266 - 0.08318 3.36 ± 0.74 +0.65−0.51

0.08318 - 0.16218 3.44 ± 0.90 +1.09−0.75

DELPHI 〈Q2〉 = 38.5 GeV2

z range F e2 /α2

0.00490 - 0.01047 11.68 ± 0.74 +1.11−0.69

0.01047 - 0.02239 8.99 ± 0.62 +0.66−0.52

0.02239 - 0.04786 6.65 ± 0.77 +0.46−0.30

0.04786 - 0.10233 5.44 ± 0.88 +0.65−0.23

0.10233 - 0.21878 3.42 ± 1.18 +0.39−0.61

DELPHI 〈Q2〉 = 62.4 GeV2

z range F e2 /α2

0.00794 - 0.01585 13.30 ± 0.99 +0.81−1.34

0.01585 - 0.03162 9.53 ± 0.84 +1.10−0.97

0.03162 - 0.06310 7.75 ± 0.86 +0.70−0.65

0.06310 - 0.12589 5.51 ± 1.02 +0.52−0.69

0.12589 - 0.25119 2.17 ± 0.75 +1.06−0.69

Table 29: Results for F e2 /α2 obtained in four intervals of Q2 with the average values of

〈Q2〉 = 15.3, 24.5, 38.5 and 62.4 GeV2 measured by the DELPHI experiment [144] at LEP2energies. The first error is statistical and the second systematic. There is no error due tomodel dependence.

115

D Results on hadronic structure of the virtual photon

In this Appendix we present the results on the photon hadronic structure function F γ2 ob-

tained by the LEP experiments OPAL, ALEPH, DELPHI and L3 as well as by the otherexperiments including AMY, CELLO, JADE, PLUTO, TASSO, TOPAZ and TPC/2γ. Inall cases the structure functions scaled by the fine structure constant are given. Dependingon the measurement, both statistical and systematical errors or only statistical or total errorare given.

PLUTO

〈Q2〉 = 5 GeV2 〈P 2〉 = 0.35 GeV2

〈x〉 x range F γeff/α ±σtot

0.05 0.00 - 0.10 0.22 ±0.04

0.19 0.10 - 0.28 0.27 ±0.07

0.39 0.28 - 0.50 0.35 ±0.11

0.63 0.50 - 0.76 0.54 ±0.25

PLUTO

〈Q2〉 = 5 GeV2

〈P 2〉 F γeff/α ±σtot

0.0 0.35 ±0.04

0.2 0.26 ±0.05

0.4 0.26 ±0.05

0.6 0.16 ±0.05

0.8 0.20 ±0.12

Table 30: Effective structure function F γeff scaled by the fine structure constant measured by

the PLUTO experiment [145]. The structure function is given as a function of x for virtualphotons at 〈P 2〉 = 0.35 GeV2, 〈Q2〉 = 5 GeV2 and as a function of P 2 for 〈Q2〉 = 5 GeV2.The point at P 2 = 0 was taken from [122]. The values given in the tables have been readout from the figures in the above publications.

L3

〈Q2〉 = 120 GeV2 〈P 2〉 = 3.7 GeV2

〈x〉 x range F γeff/α ±σstat ±σsys

0.13 0.05 - 0.20 0.42 ±0.16 ±0.05

0.30 0.20 - 0.40 0.71 ±0.24 ±0.09

0.50 0.40 - 0.60 0.72 ±0.34 ±0.09

0.70 0.60 - 0.80 1.27 ±0.51 ±0.16

0.89 0.80 - 0.98 1.48 ±0.66 ±0.19

L3

〈Q2〉 = 120 GeV2 0.05 < x < 0.98

〈P 2〉 F γeff/α ±σstat ±σsys

0.0 0.83 ±0.06 ±0.08

2.0 0.87 ±0.25 ±0.11

3.9 1.00 ±0.32 ±0.13

6.4 1.02 ±0.70 ±0.13

3.7 0.94 ±0.19 ±0.12

Table 31: Effective structure function F γeff scaled by the fine structure constant measured

by the L3 experiment [146]. The structure function is given as a function of x for virtualphotons at 〈P 2〉 = 3.7 GeV2, 〈Q2〉 = 120 GeV2 and as a function of P 2 for 〈Q2〉 = 120 GeV2,0.05 < x < 0.98. In the last row in the right table the structure function value averaged overwhole P 2 range is given.

116

OPAL

〈x〉 x range dσee/dx ±σstat ±σsys σγ⋆γ⋆ ±σstat ±σsys

[pb] [nb]

0.06 0.0 - 0.1 1.46 ±0.27 +0.11−0.26 4.66 ±0.85 +0.33

−0.83

0.15 0.1 - 0.2 1.22 ±0.20 +0.10−0.19 8.92 ±1.42 +0.71

−1.39

0.26 0.2 - 0.35 0.55 ±0.11 +0.11−0.20 5.76 ±1.10 +1.18

−2.10

〈Q2〉 Q2 range dσee/dQ2 ±σstat ±σsys σγ⋆γ⋆ ±σstat ±σsys

[GeV2] [GeV2] [pb/GeV2] [nb]

13.6 10 - 16 0.016 ±0.003 +0.002−0.003 5.63 ±1.23 +0.77

−1.05

18.9 16 - 22 0.026 ±0.004 +0.003−0.006 6.37 ±0.91 +0.75

−1.56

24.4 22 - 27 0.013 ±0.003 +0.001−0.003 3.91 ±0.89 +0.36

−0.98

〈W 〉 W range dσee/dW ±σstat ±σsys σγ⋆γ⋆ ±σstat ±σsys

[GeV2] [GeV2] [pb/GeV2] [nb]

7.2 5 - 10 0.031 ±0.004 +0.0053−0.0098 6.79 ±0.96 +1.17

−2.15

12.4 10 - 15 0.017 ±0.004 +0.0012−0.0015 7.11 ±1.51 +0.52

−0.63

20.6 15 - 35 0.005 ±0.001 +0.0004−0.0010 5.53 ±1.14 +0.38

−1.11

41.5 35 - 50 0.001 ±0.001 +0.0001−0.0001 3.10 ±2.06 +0.29

−0.43

∆φ ∆φ range dσee/d∆φ ±σstat ±σsys σγ⋆γ⋆ ±σstat ±σsys

[rad] [rad] [pb] [nb]

0.31 0.00 - 0.63 0.081 ±0.022 +0.010−0.032 3.18 ±0.84 +0.40

−1.25

0.94 0.63 - 1.26 0.098 ±0.024 +0.012−0.035 4.25 ±1.03 +0.51

−1.53

1.57 1.26 - 1.89 0.130 ±0.026 +0.013−0.036 6.84 ±1.38 +0.68

−1.88

2.20 1.89 - 2.51 0.117 ±0.026 +0.011−0.020 7.44 ±1.67 +0.70

−1.25

2.83 2.51 - 3.14 0.132 ±0.028 +0.009−0.010 9.56 ±2.02 +0.68

−0.76

dσee/dY ±σstat ±σsys σγ⋆γ⋆ ±σstat ±σsysY Y range

[pb] [nb]

0.5 0 - 1 0.059 ±0.013 +0.029−0.033 6.11 ±1.34 +2.95

−3.42

1.5 1 - 2 0.105 ±0.018 +0.007−0.021 6.84 ±1.17 +0.43

−1.39

2.5 2 - 3 0.108 ±0.020 +0.009−0.011 7.99 ±1.51 +0.69

−0.83

3.5 3 - 4 0.040 ±0.014 +0.003−0.009 3.78 ±1.38 +0.32

−0.84

5.0 4 - 6 0.026 ±0.010 +0.003−0.010 4.97 ±1.82 +0.49

−1.81

Table 32: The differential cross-section for the process e+e− → e+e− hadrons and the totalcross section for the process γ⋆γ⋆ → hadrons in the region E1,2 > 0.4Eb, 34 < θ1,2 < 55mrad and W > 5 GeV, as a function of x, Q2, W , ∆φ and Y . The average value of x, Q2,W and the central value for ∆φ and Y in a bin, value of the bin boundaries, value of thecross-section with statistical and systematic errors, are given.

117

ALEPH

etag range dσee/detag ±σstat ±σsys

[pb]

0.30 - 0.40 2.85 ±0.47 ±1.03

0.40 - 0.50 2.78 ±0.39 ±0.30

0.50 - 0.60 3.01 ±0.37 ±0.32

0.60 - 0.70 3.12 ±0.36 ±0.10

0.70 - 0.80 4.36 ±0.36 ±0.21

0.80 - 0.90 5.19 ±0.36 ±0.27

0.90 - 0.95 11.51 ±0.78 ±0.40

0.95 - 1.00 27.24 ±1.75 ±2.99

θ range dσee/dθ ±σstat ±σsys

[rad] [pb/rad]

0.035 - 0.040 110.11 ±9.63 ±3.92

0.040 - 0.045 122.18 ±9.94 ±3.01

0.045 - 0.050 94.71 ±8.01 ±2.47

0.050 - 0.055 81.16 ±7.08 ±2.73

0.060 - 0.070 40.57 ±3.63 ±1.65

0.070 - 0.090 28.08 ±2.14 ±0.92

0.090 - 0.120 15.81 ±1.29 ±0.55

0.120 - 0.155 6.49 ±0.84 ±0.27

Q2 range dσee/dQ2 ±σstat ±σsys

[GeV2] [pb/GeV2]

2 - 6 0.0133 ±0.0062 ±0.0056

6 - 10 0.0852 ±0.0112 ±0.0059

10 - 20 0.1290 ±0.0070 ±0.0052

20 - 30 0.0785 ±0.0056 ±0.0161

30 - 40 0.0415 ±0.0081 ±0.0073

40 - 50 0.0263 ±0.0030 ±0.0013

50 - 70 0.0160 ±0.0016 ±0.0004

70 - 100 0.0111 ±0.0011 ±0.0008

100 - 150 0.0038 ±0.0005 ±0.0003

150 - 200 0.0010 ±0.0003 ±0.0002

ALEPH

∆φ range dσee/d∆φ ±σstat ±σsys

[rad] [pb/rad]

0.00 - 0.31 0.45 ±0.07 ±0.02

0.31 - 0.63 0.49 ±0.07 ±0.02

0.63 - 0.94 0.58 ±0.08 ±0.06

0.94 - 1.26 0.54 ±0.08 ±0.07

1.26 - 1.57 0.53 ±0.08 ±0.06

1.57 - 1.88 0.62 ±0.08 ±0.05

1.88 - 2.20 0.82 ±0.09 ±0.03

2.20 - 2.51 0.58 ±0.09 ±0.03

2.51 - 2.83 0.83 ±0.10 ±0.08

2.83 - 3.14 0.88 ±0.13 ±0.14

W range dσee/dW ±σstat ±σsys

[GeV2] [pb/GeV2]

3 - 5 0.1409 ±0.0164 ±0.0257

5 - 10 0.1263 ±0.0095 ±0.0033

10 - 15 0.0763 ±0.0072 ±0.0032

15 - 35 0.0276 ±0.0023 ±0.0023

35 - 50 0.0102 ±0.0020 ±0.0021

50 - 80 0.0005 +0.0010−0.0005 ±0.0020

x range dσee/dx ±σstat ±σsys

[pb]

0.00 - 0.05 17.71 ±1.38 ±2.00

0.05 - 0.10 14.48 ±1.05 ±1.13

0.10 - 0.15 11.97 ±0.95 ±0.77

0.15 - 0.20 8.20 ±0.79 ±0.44

0.20 - 0.25 6.62 ±0.72 ±0.56

0.25 - 0.30 7.31 ±0.70 ±0.43

0.30 - 0.40 3.71 ±0.36 ±0.22

0.40 - 0.50 2.19 ±0.28 ±0.14

0.50 - 1.00 0.42 ±0.06 ±0.09

Table 33: The differential cross-section for the process e+e− → e+e− hadrons and the totalcross section for the process γ⋆γ⋆ → hadrons in the region E1,2 > 0.4Eb, 34 < θ1,2 < 55mrad and W > 5 GeV, as a function of x, Q2, W , ∆φ and Y . The average value of x, Q2,W and the central value for ∆φ and Y in a bin, value of the bin boundaries, value of thecross-section with statistical and systematic errors, are given.

118

ALEPH

dσee/dY ±σstat ±σsys dσee/dY ±σstat ±σsys σγ⋆γ⋆ ±σstat ±σsysY range

[pb] [pb] [nb]

-1.0 - 0.0 0.224 ±0.028 ±0.031 0.142 ±0.022 ±0.015 2.0 ±0.3 ±0.2

0.0 - 1.0 0.437 ±0.039 ±0.015 0.321 ±0.034 ±0.014 4.5 ±0.5 ±0.2

1.0 - 2.0 0.542 ±0.045 ±0.022 0.385 ±0.038 ±0.014 6.9 ±0.7 ±0.3

2.0 - 3.0 0.422 ±0.040 ±0.030 0.232 ±0.030 ±0.019 4.7 ±0.6 ±0.4

3.0 - 4.0 0.227 ±0.035 ±0.017 0.115 ±0.025 ±0.008 3.1 ±0.7 ±0.2

4.0 - 5.0 0.117 ±0.026 ±0.024 0.073 ±0.021 ±0.016 3.5 ±1.0 ±0.8

5.0 - 6.0 0.026 +0.040−0.020 ±0.013 0.012 +0.028

−0.012 ±0.010 1.5 +3.5−1.5 ±1.3

6.0 - 7.0 0.008 +0.030−0.008 ±0.005 0.005 +0.024

−0.005 ±0.004 2.5 +12.−2.5 ±2.0

Table 34: The differential cross-section for the process e+e− → e+e− hadrons and the totalcross section for the process γ⋆γ⋆ → hadrons in the region E1,2 > 0.4Eb, 34 < θ1,2 < 55mrad and W > 5 GeV, as a function of x, Q2, W , ∆φ and Y . The average value of x, Q2,W and the central value for ∆φ and Y in a bin, value of the bin boundaries, value of thecross-section with statistical and systematic errors, are given.

L3

〈Q2〉 Q2 range dσee/dQ2 ±σstat ±σsys ±σrad σγ⋆γ⋆ ±σstat ±σsys ±σrad

[GeV2] [GeV2] [pb/GeV2] [nb]

12.0 10 - 14 0.0718 ±0.0070 ±0.0061 ±0.0022 6.49 ±0.64 ±0.55 ±0.20

15.9 14 - 18 0.0522 ±0.0057 ±0.0044 ±0.0016 4.84 ±0.53 ±0.41 ±0.15

20.5 18 - 24 0.0273 ±0.0033 ±0.0023 ±0.0008 4.54 ±0.55 ±0.39 ±0.14

27.0 24 - 32 0.0066 ±0.0014 ±0.0006 ±0.0002 3.38 ±0.74 ±0.29 ±0.10

〈W 〉 W range dσee/dW ±σstat ±σsys ±σrad σγ⋆γ⋆ ±σstat ±σsys ±σrad

[GeV2] [GeV2] [pb/GeV2] [nb]

7.2 5 - 10 0.0747 ±0.0096 ±0.0063 ±0.0023 6.34 ±0.82 ±0.54 ±0.19

13.9 10 - 20 0.0263 ±0.0024 ±0.0022 ±0.0008 5.27 ±0.49 ±0.45 ±0.16

27.9 20 - 40 0.0062 ±0.0007 ±0.0005 ±0.0003 3.71 ±0.40 ±0.32 ±0.16

61.6 40 - 100 0.0014 ±0.0002 ±0.0001 ±0.0001 5.24 ±0.79 ±0.45 ±0.34

Y Y range dσee/dY ±σstat ±σsys ±σrad σγ⋆γ⋆ ±σstat ±σsys ±σrad

[pb] [nb]

2.2 2.0 - 2.5 0.315 ±0.048 ±0.027 ±0.009 5.65 ±0.86 ±0.48 ±0.17

2.9 2.5 - 3.5 0.184 ±0.018 ±0.016 ±0.006 4.90 ±0.48 ±0.42 ±0.16

4.2 3.5 - 5.0 0.085 ±0.009 ±0.007 ±0.004 3.99 ±0.42 ±0.34 ±0.19

5.9 5.0 - 7.0 0.037 ±0.006 ±0.003 ±0.002 5.82 ±0.97 ±0.49 ±0.37

Table 35: The differential cross-section for the process e+e− → e+e− hadrons and the totalcross section for the process γ⋆γ⋆ → hadrons in the region E1,2 > 0.4Eb, 34 < θ1,2 < 55mrad and W > 5 GeV, as a function of x, Q2, W , ∆φ and Y . The average value of x, Q2,W and the central value for ∆φ and Y in a bin, value of the bin boundaries, value of thecross-section with statistical and systematic errors, are given.

119

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